p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.67C23, (C2×C4)⋊3Q8, (C2×Q8)⋊3C4, C2.5(C4×Q8), (C2×C4).68D4, C2.3(C4⋊Q8), C4.7(C22⋊C4), (C2×C42).11C2, C22.40(C2×D4), C2.5(C22⋊Q8), (C22×Q8).1C2, C22.15(C2×Q8), C2.4(C4.4D4), (C22×C4).6C22, C22.25(C4○D4), C2.C42.7C2, C22.40(C22×C4), (C2×C4⋊C4).8C2, (C2×C4).19(C2×C4), C2.9(C2×C22⋊C4), SmallGroup(64,72)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.67C23
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=c, e2=abc, f2=b, ab=ba, ac=ca, ede-1=ad=da, ae=ea, af=fa, bc=cb, fdf-1=bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ef=fe >
Subgroups: 137 in 93 conjugacy classes, 53 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, C23.67C23
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C23.67C23
Character table of C23.67C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | i | -i | 1 | -1 | 1 | -i | i | -i | i | -i | i | -i | -1 | 1 | i | -i | i | 1 | -1 | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | -i | -1 | 1 | -1 | i | -i | i | i | -i | i | i | 1 | 1 | i | -i | -i | -1 | -1 | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | i | -i | 1 | -1 | 1 | -i | i | -i | i | -i | i | i | 1 | -1 | -i | i | -i | -1 | 1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | -i | -1 | 1 | -1 | i | -i | i | i | -i | i | -i | -1 | -1 | -i | i | i | 1 | 1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | i | -1 | 1 | -1 | -i | i | -i | -i | i | -i | i | -1 | -1 | i | -i | -i | 1 | 1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -i | i | 1 | -1 | 1 | i | -i | i | -i | i | -i | -i | 1 | -1 | i | -i | i | -1 | 1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | i | -1 | 1 | -1 | -i | i | -i | -i | i | -i | -i | 1 | 1 | -i | i | i | -1 | -1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -i | i | 1 | -1 | 1 | i | -i | i | -i | i | -i | i | -1 | 1 | -i | i | -i | 1 | -1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ22 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ23 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ25 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 2i | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | -2i | 0 | 0 | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►Regular action on 64 points
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 62)(6 63)(7 64)(8 61)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 39)(34 40)(35 37)(36 38)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 29 21 45)(2 58 22 18)(3 31 23 47)(4 60 24 20)(5 16 34 56)(6 41 35 25)(7 14 36 54)(8 43 33 27)(9 57 49 17)(10 30 50 46)(11 59 51 19)(12 32 52 48)(13 61 53 39)(15 63 55 37)(26 40 42 62)(28 38 44 64)
(1 55 51 41)(2 42 52 56)(3 53 49 43)(4 44 50 54)(5 58 62 48)(6 45 63 59)(7 60 64 46)(8 47 61 57)(9 27 23 13)(10 14 24 28)(11 25 21 15)(12 16 22 26)(17 33 31 39)(18 40 32 34)(19 35 29 37)(20 38 30 36)
G:=sub<Sym(64)| (1,9)(2,10)(3,11)(4,12)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,62)(6,63)(7,64)(8,61)(9,23)(10,24)(11,21)(12,22)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(33,39)(34,40)(35,37)(36,38)(41,55)(42,56)(43,53)(44,54)(45,59)(46,60)(47,57)(48,58), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,29,21,45)(2,58,22,18)(3,31,23,47)(4,60,24,20)(5,16,34,56)(6,41,35,25)(7,14,36,54)(8,43,33,27)(9,57,49,17)(10,30,50,46)(11,59,51,19)(12,32,52,48)(13,61,53,39)(15,63,55,37)(26,40,42,62)(28,38,44,64), (1,55,51,41)(2,42,52,56)(3,53,49,43)(4,44,50,54)(5,58,62,48)(6,45,63,59)(7,60,64,46)(8,47,61,57)(9,27,23,13)(10,14,24,28)(11,25,21,15)(12,16,22,26)(17,33,31,39)(18,40,32,34)(19,35,29,37)(20,38,30,36)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,62)(6,63)(7,64)(8,61)(9,23)(10,24)(11,21)(12,22)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(33,39)(34,40)(35,37)(36,38)(41,55)(42,56)(43,53)(44,54)(45,59)(46,60)(47,57)(48,58), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,29,21,45)(2,58,22,18)(3,31,23,47)(4,60,24,20)(5,16,34,56)(6,41,35,25)(7,14,36,54)(8,43,33,27)(9,57,49,17)(10,30,50,46)(11,59,51,19)(12,32,52,48)(13,61,53,39)(15,63,55,37)(26,40,42,62)(28,38,44,64), (1,55,51,41)(2,42,52,56)(3,53,49,43)(4,44,50,54)(5,58,62,48)(6,45,63,59)(7,60,64,46)(8,47,61,57)(9,27,23,13)(10,14,24,28)(11,25,21,15)(12,16,22,26)(17,33,31,39)(18,40,32,34)(19,35,29,37)(20,38,30,36) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,38),(6,39),(7,40),(8,37),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,63),(34,64),(35,61),(36,62)], [(1,51),(2,52),(3,49),(4,50),(5,62),(6,63),(7,64),(8,61),(9,23),(10,24),(11,21),(12,22),(13,27),(14,28),(15,25),(16,26),(17,31),(18,32),(19,29),(20,30),(33,39),(34,40),(35,37),(36,38),(41,55),(42,56),(43,53),(44,54),(45,59),(46,60),(47,57),(48,58)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,29,21,45),(2,58,22,18),(3,31,23,47),(4,60,24,20),(5,16,34,56),(6,41,35,25),(7,14,36,54),(8,43,33,27),(9,57,49,17),(10,30,50,46),(11,59,51,19),(12,32,52,48),(13,61,53,39),(15,63,55,37),(26,40,42,62),(28,38,44,64)], [(1,55,51,41),(2,42,52,56),(3,53,49,43),(4,44,50,54),(5,58,62,48),(6,45,63,59),(7,60,64,46),(8,47,61,57),(9,27,23,13),(10,14,24,28),(11,25,21,15),(12,16,22,26),(17,33,31,39),(18,40,32,34),(19,35,29,37),(20,38,30,36)]])
C23.67C23 is a maximal subgroup of
(C2×Q8)⋊C8 (C2×C42).C4 (C2×Q8).Q8 2- 1+4⋊2C4 C4.10D4⋊2C4 (C2×SD16)⋊14C4 (C2×C4)⋊9Q16 C8.C22⋊C4 C4.68(C4×D4) C2.(C4×Q16) C2.(C8⋊8D4) C2.(C8⋊D4) (C2×C4)⋊9SD16 (C2×C4)⋊6Q16 (C2×Q16)⋊10C4 C8⋊(C22⋊C4) M4(2)⋊4D4 (C2×C8)⋊20D4 (C2×C8).41D4 (C2×C4)⋊2Q16 (C2×Q8)⋊Q8 C4⋊C4.85D4 (C2×Q8)⋊2Q8 M4(2).7D4 (C2×C4)⋊5SD16 (C2×C4)⋊3Q16 (C2×Q8).8Q8 (C2×C8).52D4 (C2×C4).19Q16 (C2×Q8).109D4 (C2×C8).60D4 (C2×C8).170D4 (C2×C8).171D4 C23.179C24 C4×C22⋊Q8 C4×C4.4D4 C4×C4⋊Q8 C23.192C24 C24.542C23 C23.202C24 C42.159D4 C42.160D4 C42.161D4 C23.211C24 C23.214C24 C24.205C23 C24.549C23 Q8×C22⋊C4 C23.238C24 C24.558C23 C23.244C24 C23.247C24 C24.220C23 C23.250C24 C24.221C23 C24.227C23 C23.261C24 C23.263C24 C24.244C23 C23.309C24 C23.315C24 C24.252C23 C23.321C24 C23.323C24 C23.327C24 C23.329C24 C24.263C23 C24.264C23 C23.334C24 C24.565C23 C24.567C23 C24.267C23 C23.346C24 C24.271C23 C23.348C24 C23.350C24 C23.351C24 C23.353C24 C23.359C24 C24.285C23 C23.374C24 C23.377C24 C23.388C24 C24.301C23 C23.392C24 C24.308C23 C23.402C24 C24.579C23 C23.411C24 C23.414C24 C23.420C24 C24.311C23 C24.313C23 C24.315C23 C23.432C24 C42⋊21D4 C42.168D4 C42.169D4 C42.170D4 C23.449C24 C42⋊6Q8 C23.456C24 C23.457C24 C24.332C23 C42.174D4 C42.176D4 C42.177D4 C23.472C24 C24.338C23 C42.179D4 C42.180D4 C23.486C24 C23.488C24 C24.346C23 C42.182D4 C42⋊8Q8 C24.355C23 C42⋊9Q8 C42⋊28D4 C42.186D4 C23.525C24 C42.193D4 C42.195D4 C23.545C24 C23.550C24 C23.553C24 C24.379C23 C42⋊11Q8 C24.394C23 C23.589C24 C23.590C24 C23.592C24 C24.405C23 C23.600C24 C23.602C24 C24.408C23 C24.412C23 C23.612C24 C23.613C24 C23.615C24 C23.616C24 C24.420C23 C24.421C23 C23.630C24 C23.631C24 C23.632C24 C23.633C24 C23.634C24 C23.645C24 C23.651C24 C23.654C24 C23.655C24 C23.658C24 C23.659C24 C23.662C24 C23.663C24 C23.664C24 C23.674C24 C23.675C24 C23.688C24 C23.689C24 C23.698C24 C23.699C24 C24.456C23 C23.705C24 C23.706C24 C23.708C24 C23.709C24 C23.711C24 C23.731C24 C23.732C24 C23.733C24 C42⋊46D4 C24.599C23 C42.440D4 C43⋊14C2 C42⋊18Q8 C42⋊15Q8
C2p.(C4×Q8): C42⋊14Q8 C42⋊4Q8 C23.237C24 C23.251C24 (C2×C12)⋊Q8 (C2×Dic6)⋊7C4 C4.(D6⋊C4) (C6×Q8)⋊7C4 ...
C23.67C23 is a maximal quotient of
C24.625C23 C24.626C23 C24.631C23 C24.635C23 C24.636C23 C24.176C23 M4(2)⋊8Q8 C42.128D4
(C2×C4p)⋊Q8: C42.327D4 C42.120D4 C42.436D4 C42.125D4 (C2×C12)⋊Q8 (C2×Dic6)⋊7C4 (C2×C20)⋊Q8 (C2×C20)⋊10Q8 ...
C2p.(C4×Q8): M4(2)⋊7Q8 C42.121D4 C42.122D4 C42.123D4 C42.437D4 C42.124D4 C42⋊16Q8 C42⋊Q8 ...
Matrix representation of C23.67C23 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,2] >;
C23.67C23 in GAP, Magma, Sage, TeX
C_2^3._{67}C_2^3 % in TeX
G:=Group("C2^3.67C2^3"); // GroupNames label
G:=SmallGroup(64,72);
// by ID
G=gap.SmallGroup(64,72);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,343,362,86]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=c,e^2=a*b*c,f^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,e*f=f*e>;
// generators/relations
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