G_1: |G|=128, smooth pairs 256, orbits 1, orders {1: 1, 2: 19, 4: 60, 8: 48}
G_2: |G|=128, smooth pairs 256, orbits 1, orders {1: 1, 2: 27, 4: 52, 8: 48}
G_3: |G|=128, smooth pairs 512, orbits 1, orders {1: 1, 2: 31, 4: 64, 8: 32}
G_4: |G|=128, smooth pairs 512, orbits 1, orders {1: 1, 2: 27, 4: 52, 8: 48}
LMFDB [128,75]: degree 128, |G|=128, smooth pairs 512, orbits 1, orders {1: 1, 2: 31, 4: 64, 8: 32} -> matches G_[3]
LMFDB [128,134]: degree 128, |G|=128, smooth pairs 256, orbits 1, orders {1: 1, 2: 27, 4: 52, 8: 48} -> matches G_[2]
LMFDB [128,136]: degree 128, |G|=128, smooth pairs 256, orbits 1, orders {1: 1, 2: 19, 4: 60, 8: 48} -> matches G_[1]
LMFDB [128,138]: degree 128, |G|=128, smooth pairs 512, orbits 1, orders {1: 1, 2: 27, 4: 52, 8: 48} -> matches G_[4]
all 32 vectors: group order 128, orders (2,4,8), product identity: True
Counter({('[128,75]', True, True): 16, ('[128,138]', True, True): 8, ('[128,134]', True, True): 4, ('[128,136]', True, True): 4})
