p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.65C23, C4⋊C4⋊6C4, C4⋊2(C4⋊C4), C2.8(C4×D4), C2.4(C4×Q8), C2.2(C4⋊Q8), (C2×C4).15Q8, (C2×C4).116D4, C2.4(C4⋊D4), (C2×C42).10C2, C22.38(C2×D4), C2.4(C22⋊Q8), C22.14(C2×Q8), C2.3(C42.C2), C22.23(C4○D4), C2.C42.6C2, (C22×C4).24C22, C22.38(C22×C4), C2.8(C2×C4⋊C4), (C2×C4⋊C4).7C2, (C2×C4).18(C2×C4), SmallGroup(64,70)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.65C23
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=e2=c, 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: 121 in 85 conjugacy classes, 53 normal (23 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, C23, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C23.65C23
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C23.65C23
Character table of C23.65C23
| 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 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 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 | orthogonal lifted from D4 |
| ρ21 | 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 | symplectic lifted from Q8, Schur index 2 |
| ρ22 | 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 | symplectic lifted from Q8, Schur index 2 |
| ρ23 | 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 |
| 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 | 2i | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 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 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | 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 47 3 45)(2 20 4 18)(5 16 7 14)(6 41 8 43)(9 19 11 17)(10 48 12 46)(13 37 15 39)(21 31 23 29)(22 60 24 58)(25 33 27 35)(26 64 28 62)(30 50 32 52)(34 56 36 54)(38 44 40 42)(49 59 51 57)(53 63 55 61)
(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), (1,47,3,45)(2,20,4,18)(5,16,7,14)(6,41,8,43)(9,19,11,17)(10,48,12,46)(13,37,15,39)(21,31,23,29)(22,60,24,58)(25,33,27,35)(26,64,28,62)(30,50,32,52)(34,56,36,54)(38,44,40,42)(49,59,51,57)(53,63,55,61), (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), (1,47,3,45)(2,20,4,18)(5,16,7,14)(6,41,8,43)(9,19,11,17)(10,48,12,46)(13,37,15,39)(21,31,23,29)(22,60,24,58)(25,33,27,35)(26,64,28,62)(30,50,32,52)(34,56,36,54)(38,44,40,42)(49,59,51,57)(53,63,55,61), (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)], [(1,47,3,45),(2,20,4,18),(5,16,7,14),(6,41,8,43),(9,19,11,17),(10,48,12,46),(13,37,15,39),(21,31,23,29),(22,60,24,58),(25,33,27,35),(26,64,28,62),(30,50,32,52),(34,56,36,54),(38,44,40,42),(49,59,51,57),(53,63,55,61)], [(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)]])
C23.65C23 is a maximal subgroup of
C4⋊C4⋊C8 C2.D8⋊4C4 C4.Q8⋊9C4 C4.Q8⋊10C4 C2.D8⋊5C4 C4⋊C4⋊7D4 C4⋊C4.94D4 C4⋊C4.95D4 C4⋊C4⋊Q8 (C2×C8)⋊Q8 C2.(C8⋊Q8) C4⋊C4.106D4 (C2×Q8).8Q8 (C2×C4).23D8 (C2×C8).52D4 (C2×C8).1Q8 C2.(C8⋊3Q8) (C2×C8).24Q8 (C2×C4).26D8 (C2×C4).21Q16 C4.(C4⋊Q8) (C2×C4).28D8 (C2×C4).23Q16 C4⋊C4.Q8 C42⋊14Q8 C23.178C24 C4×C4⋊D4 C4×C42.C2 C24.545C23 C23.199C24 C23.201C24 C23.202C24 C24.195C23 C23.211C24 C42.33Q8 C42⋊4Q8 C24.203C23 C24.204C23 C23.218C24 C23.226C24 C23.227C24 C24.208C23 C23.229C24 D4×C4⋊C4 C23.231C24 Q8×C4⋊C4 C23.233C24 C23.241C24 C24.215C23 C23.251C24 C23.252C24 C23.255C24 C24.225C23 C24.227C23 C23.264C24 C24.230C23 C23.313C24 C24.249C23 C23.315C24 C23.316C24 C24.252C23 C24.254C23 C23.321C24 C23.322C24 C23.323C24 C24.258C23 C24.259C23 C23.327C24 C23.328C24 C23.329C24 C24.567C23 C24.267C23 C24.568C23 C24.268C23 C24.569C23 C23.345C24 C23.346C24 C24.271C23 C23.348C24 C23.349C24 C23.351C24 C23.352C24 C23.353C24 C23.354C24 C24.276C23 C23.362C24 C24.285C23 C24.286C23 C23.369C24 C24.289C23 C24.572C23 C23.375C24 C24.295C23 C23.379C24 C23.385C24 C24.299C23 C24.300C23 C24.304C23 C23.395C24 C23.396C24 C23.397C24 C23.405C24 C23.406C24 C23.407C24 C23.408C24 C23.409C24 C23.411C24 C23.412C24 C23.413C24 C23.414C24 C23.418C24 C23.419C24 C23.422C24 C23.425C24 C23.426C24 C23.428C24 C23.429C24 C23.430C24 C23.433C24 C42.166D4 C42.169D4 C23.449C24 C42⋊6Q8 C42⋊7Q8 C42.35Q8 C24.326C23 C23.456C24 C23.458C24 C24.331C23 C42.176D4 C42.36Q8 C42.37Q8 C23.473C24 C42.178D4 C42.179D4 C42.180D4 C23.483C24 C42.181D4 C23.485C24 C23.486C24 C24.345C23 C23.488C24 C24.346C23 C23.490C24 C23.494C24 C23.496C24 C42⋊23D4 C23.502C24 C42.184D4 C42.38Q8 C23.508C24 C42.185D4 C42⋊9Q8 C23.524C24 C23.525C24 C42.187D4 C42.188D4 C23.544C24 C23.545C24 C42.39Q8 C24.375C23 C24.376C23 C23.554C24 C23.555C24 C24.378C23 C42.198D4 C42⋊11Q8 C23.567C24 C23.572C24 C23.574C24 C23.581C24 C24.389C23 C23.583C24 C23.589C24 C24.401C23 C23.595C24 C24.405C23 C24.407C23 C23.602C24 C24.408C23 C23.605C24 C23.607C24 C24.412C23 C23.613C24 C23.615C24 C23.618C24 C23.619C24 C23.620C24 C23.621C24 C23.622C24 C24.418C23 C23.624C24 C23.625C24 C23.626C24 C23.627C24 C24.421C23 C23.632C24 C23.634C24 C24.426C23 C24.427C23 C24.428C23 C23.655C24 C24.438C23 C23.658C24 C24.443C23 C23.666C24 C23.667C24 C23.668C24 C23.669C24 C24.445C23 C23.671C24 C23.672C24 C23.673C24 C23.674C24 C23.675C24 C23.676C24 C23.677C24 C23.679C24 C24.448C23 C23.683C24 C23.687C24 C23.688C24 C24.454C23 C23.691C24 C23.692C24 C23.696C24 C23.698C24 C23.700C24 C23.701C24 C23.702C24 C23.703C24 C23.705C24 C23.706C24 C23.707C24 C23.708C24 C23.709C24 C23.710C24 C23.736C24 C23.737C24 C23.738C24 C23.739C24 C42.439D4 C24.598C23 C24.599C23 C42.440D4 C43.15C2 C43⋊13C2 C42⋊15Q8 C43.18C2
C4p⋊(C4⋊C4): C8⋊7(C4⋊C4) C8⋊5(C4⋊C4) C4.(C4×Q8) C8⋊(C4⋊C4) C12⋊4(C4⋊C4) C12⋊(C4⋊C4) C4⋊C4⋊6Dic3 C20⋊7(C4⋊C4) ...
C2p.(C4×D4): C2.(C4×D8) Q8⋊(C4⋊C4) D4⋊(C4⋊C4) Q8⋊C4⋊C4 C2.(C8⋊8D4) C2.(C8⋊7D4) C2.(C8⋊D4) C2.(C8⋊2D4) ...
C23.65C23 is a maximal quotient of
C24.625C23 C24.626C23 C24.631C23 C24.632C23 C24.634C23 C42.61Q8 C42.27Q8 M4(2).5Q8 M4(2).6Q8 M4(2).27D4
C4p⋊(C4⋊C4): C8⋊7(C4⋊C4) C8⋊5(C4⋊C4) C4.(C4×Q8) C8⋊(C4⋊C4) C12⋊4(C4⋊C4) C12⋊(C4⋊C4) C4⋊C4⋊6Dic3 C20⋊7(C4⋊C4) ...
C2p.(C4×Q8): C42.62Q8 C42.28Q8 C42.29Q8 C42.30Q8 C42.31Q8 C42.430D4 C6.(C4×Q8) Dic3⋊(C4⋊C4) ...
Matrix representation of C23.65C23 ►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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 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 | 3 | 2 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,3,0,0,0,0,0,2] >;
C23.65C23 in GAP, Magma, Sage, TeX
C_2^3._{65}C_2^3 % in TeX
G:=Group("C2^3.65C2^3"); // GroupNames label
G:=SmallGroup(64,70);
// by ID
G=gap.SmallGroup(64,70);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,192,121,199,362,86]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=e^2=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
Export