parsed triangle-group actions: 7941
(2,4,8) actions of order 16(g-1), g-1 power of 2: 35
(2,3,8) actions of order 48(g-1) (all g): [(2, 'Action is reflexible [m]'), (3, 'Action is reflexible [m]'), (5, 'Action is reflexible [m]'), (8, 'Action is reflexible [m] and there is a non-orientable quotient action [m]'), (8, 'Action is reflexible [m] and there is a non-orientable quotient action [m]'), (10, 'Action is chiral'), (16, 'Action is reflexible [m] and there is a non-orientable quotient action [m]'), (17, 'Action is reflexible [m]'), (22, 'Action is reflexible [m] and there is a non-orientable quotient action [m]'), (22, 'Action is reflexible [m]'), (28, 'Action is chiral'), (33, 'Action is reflexible [m]'), (37, 'Action is chiral'), (46, 'Action is chiral'), (46, 'Action is reflexible [m]'), (46, 'Action is reflexible [m] and there is a non-orientable quotient action [m]'), (52, 'Action is reflexible [m]'), (52, 'Action is reflexible [m]'), (65, 'Action is reflexible [m]'), (82, 'Action is chiral'), (82, 'Action is reflexible [m]'), (85, 'Action is reflexible [m] and there is a non-orientable quotient action [m]'), (85, 'Action is reflexible [m]')]
all (2,4,8) entries: genera/orders [(2, 16), (3, 32), (5, 64), (9, 128), (10, 144), (11, 160), (17, 256), (19, 288), (21, 320), (22, 336), (27, 416), (28, 432), (33, 512), (35, 544), (37, 576), (41, 640), (43, 672), (46, 720), (51, 800), (53, 832), (55, 864), (59, 928), (65, 1024), (69, 1088), (73, 1152), (75, 1184), (81, 1280), (82, 1296), (83, 1312), (85, 1344), (91, 1440), (99, 1568), (101, 1600)]
(2,4,8) entries lacking the standard relators G.1^2, G.2^4, G.3^-1*G.2^-1*G.1 (lines): []
(2,4,8) entries lacking G.3^8 (lines): [113, 222, 226, 573]
g=2 line=113 cosets=16 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=16 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[] |F/ncl|=48 normalF=True (0.3s)
g=3 line=222 cosets=32 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=32 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[] |F/ncl|=96 normalF=True (0.3s)
g=3 line=226 cosets=32 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=32 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[3] |F/ncl|=48 normalF=False (0.4s)
g=5 line=573 cosets=64 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=64 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[3] |F/ncl|=24 normalF=False (0.6s)
g=5 line=578 cosets=64 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=64 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[] |F/ncl|=192 normalF=True (0.9s)
g=9 line=1427 cosets=128 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=128 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[3] |F/ncl|=24 normalF=False (0.8s)
g=9 line=1432 cosets=128 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=128 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[3] |F/ncl|=24 normalF=False (0.5s)
g=9 line=1437 cosets=128 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=128 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4] |F/ncl|=192 normalF=False (1.4s)
g=9 line=1442 cosets=128 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=128 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[3, 5] |F/ncl|=24 normalF=False (1.1s)
g=17 line=4645 cosets=256 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=256 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4] |F/ncl|=192 normalF=False (2.2s)
g=17 line=4651 cosets=256 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=256 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[] |F/ncl|=768 normalF=True (4.8s)
g=17 line=4656 cosets=256 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=256 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4, 5] |F/ncl|=192 normalF=False (1.7s)
g=17 line=4661 cosets=256 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=256 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4, 5] |F/ncl|=24 normalF=False (1.7s)
g=33 line=15421 cosets=512 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=512 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4, 5] |F/ncl|=192 normalF=False (4.2s)
g=33 line=15427 cosets=512 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=512 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4, 5] |F/ncl|=192 normalF=False (3.3s)
g=33 line=15432 cosets=512 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=512 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4] |F/ncl|=768 normalF=False (6.2s)
g=33 line=15437 cosets=512 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=512 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4] |F/ncl|=768 normalF=False (8.2s)
g=33 line=15443 cosets=512 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=512 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[] |F/ncl|=1536 normalF=True (20.6s)
g=33 line=15450 cosets=512 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=512 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4] |F/ncl|=192 normalF=False (2.0s)
g=33 line=15455 cosets=512 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=512 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4] |F/ncl|=192 normalF=False (1.6s)
g=33 line=15460 cosets=512 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=512 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4, 5] |F/ncl|=24 normalF=False (3.3s)
g=65 line=53909 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is chiral' computed=False cert=[4, 6] |F/ncl|=192 normalF=False (8.1s)
g=65 line=53915 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is chiral' computed=False cert=[4, 5] |F/ncl|=1536 normalF=False (18.5s)
g=65 line=53921 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4, 5] |F/ncl|=192 normalF=False (7.2s)
g=65 line=53927 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4, 5] |F/ncl|=192 normalF=False (10.3s)
g=65 line=53934 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4] |F/ncl|=768 normalF=False (15.0s)
g=65 line=53940 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4, 5] |F/ncl|=768 normalF=False (11.7s)
g=65 line=53946 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[5] |F/ncl|=1536 normalF=False (28.7s)
g=65 line=53952 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[5] |F/ncl|=768 normalF=False (10.3s)
g=65 line=53958 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4] |F/ncl|=768 normalF=False (7.0s)
g=65 line=53964 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4, 6] |F/ncl|=768 normalF=False (17.9s)
g=65 line=53971 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4, 5] |F/ncl|=768 normalF=False (11.7s)
g=65 line=53977 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[] |F/ncl|=3072 normalF=True (48.8s)
g=65 line=53984 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4] |F/ncl|=768 normalF=False (12.7s)
g=65 line=53991 cosets=1024 ords={1: 2, 2: 4, 3: 8} x=yz:True perm_order=1024 trans=True reflexible: census='Action is reflexible [m]' computed=True cert=[4, 5] |F/ncl|=24 normalF=False (8.2s)
genus 2: kernels equal (relator test) []; rho-related []; distinct canonical forms 1/1; canon-vs-rho-canon equalities across list: 1 (expected = number reflexible); reflexible by canon: [True]
genus 3: kernels equal (relator test) []; rho-related []; distinct canonical forms 2/2; canon-vs-rho-canon equalities across list: 2 (expected = number reflexible); reflexible by canon: [True, True]
genus 5: kernels equal (relator test) []; rho-related []; distinct canonical forms 2/2; canon-vs-rho-canon equalities across list: 2 (expected = number reflexible); reflexible by canon: [True, True]
genus 9: kernels equal (relator test) []; rho-related []; distinct canonical forms 4/4; canon-vs-rho-canon equalities across list: 4 (expected = number reflexible); reflexible by canon: [True, True, True, True]
genus 17: kernels equal (relator test) []; rho-related []; distinct canonical forms 4/4; canon-vs-rho-canon equalities across list: 4 (expected = number reflexible); reflexible by canon: [True, True, True, True]
genus 33: kernels equal (relator test) []; rho-related []; distinct canonical forms 8/8; canon-vs-rho-canon equalities across list: 8 (expected = number reflexible); reflexible by canon: [True, True, True, True, True, True, True, True]
genus 65: kernels equal (relator test) []; rho-related []; distinct canonical forms 14/14; canon-vs-rho-canon equalities across list: 12 (expected = number reflexible); reflexible by canon: [False, False, True, True, True, True, True, True, True, True, True, True, True, True]

MY TABLE 1
g  16(g-1)  actions  chiral  normal_in_F  surfaces   | (2,3,8) actions of order 48(g-1)
  2      16        1       0            1         0   | 1
  3      32        2       0            1         1   | 1
  5      64        2       0            1         1   | 1
  9     128        4       0            0         4   | 0
 17     256        4       0            1         3   | 1
 33     512        8       0            1         7   | 1
 65    1024       14       2            1        15   | 1
G_1 (line 1427): {'nilpotency_class': 5, 'center': 2, 'exponent': 8, 'abelian_invariants': [2, 4], 'order_counts': {1: 1, 2: 19, 4: 60, 8: 48}, 'derived_order': 16}
G_2 (line 1432): {'nilpotency_class': 5, 'center': 2, 'exponent': 8, 'abelian_invariants': [2, 4], 'order_counts': {1: 1, 2: 27, 4: 52, 8: 48}, 'derived_order': 16}
G_3 (line 1437): {'nilpotency_class': 4, 'center': 4, 'exponent': 8, 'abelian_invariants': [2, 4], 'order_counts': {1: 1, 2: 31, 4: 64, 8: 32}, 'derived_order': 16}
G_4 (line 1442): {'nilpotency_class': 5, 'center': 2, 'exponent': 8, 'abelian_invariants': [2, 4], 'order_counts': {1: 1, 2: 27, 4: 52, 8: 48}, 'derived_order': 16}

ISOMORPHISMS among the genus-9 groups (epimorphism G_i -> G_j via a (4,8)-pair of G_j satisfying the relators of G_i):
iso matrix (i,j) -> G_i iso G_j: {'11': True, '12': False, '13': False, '14': False, '21': False, '22': True, '23': False, '24': False, '31': False, '32': False, '33': True, '34': False, '41': False, '42': False, '43': False, '44': True}
number of generating (4,8)-pairs with product of order 2: {1: 256, 2: 256, 3: 512, 4: 512}

KERNELS of L with quotient isomorphic to G_k (distinct canonical forms over all generating pairs), and their reflexibility:
G_1: 1 kernels; reflexible flags [True]; present in census list: [True]
G_2: 1 kernels; reflexible flags [True]; present in census list: [True]
G_3: 1 kernels; reflexible flags [True]; present in census list: [True]
G_4: 1 kernels; reflexible flags [True]; present in census list: [True]
total time 283.9 s
