p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C23.714C24, C24.460C23, C22.3722- 1+4, C22.4872+ 1+4, C23⋊Q8⋊60C2, C23.224(C2×D4), (C22×C4).410D4, C23.Q8⋊92C2, C2.61(C23⋊3D4), (C23×C4).499C22, (C22×C4).225C23, C23.7Q8⋊114C2, C22.446(C22×D4), C23.11D4⋊127C2, C23.10D4.71C2, (C22×D4).292C22, (C22×Q8).231C22, C23.78C23⋊62C2, C23.81C23⋊132C2, C2.C42.417C22, C2.57(C22.57C24), C2.49(C22.56C24), C2.57(C22.31C24), C2.66(C23.38C23), (C2×C4).431(C2×D4), (C2×C22⋊Q8)⋊47C2, (C2×C4⋊C4).523C22, (C2×C22⋊C4).333C22, (C2×C22.D4).31C2, SmallGroup(128,1546)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.714C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=f2=b, gag=ab=ba, ac=ca, ad=da, ae=ea, faf-1=abc, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg=ce=ec, cf=fc, cg=gc, de=ed, gfg=df=fd, dg=gd >
Subgroups: 500 in 239 conjugacy classes, 92 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C22.D4, C23×C4, C22×D4, C22×Q8, C23.7Q8, C23⋊Q8, C23.10D4, C23.78C23, C23.Q8, C23.11D4, C23.11D4, C23.81C23, C23.81C23, C2×C22⋊Q8, C2×C22.D4, C23.714C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, 2- 1+4, C23⋊3D4, C23.38C23, C22.31C24, C22.56C24, C22.57C24, C23.714C24
Character table of C23.714C24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 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 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 2 | -2 | 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 | -2 | 0 | -2 | 2 | -2 | 2 | 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 | 2 | 0 | 2 | -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 | -2 | 2 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | 4 | -4 | -4 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 4 | -4 | 4 | -4 | -4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | -4 | -4 | 4 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ24 | 4 | -4 | -4 | 4 | -4 | -4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ25 | 4 | 4 | -4 | -4 | -4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ26 | 4 | 4 | 4 | 4 | -4 | -4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
►On 64 points
Generators in S64
(1 41)(2 42)(3 43)(4 44)(5 33)(6 34)(7 35)(8 36)(9 13)(10 14)(11 15)(12 16)(17 51)(18 52)(19 49)(20 50)(21 47)(22 48)(23 45)(24 46)(25 29)(26 30)(27 31)(28 32)(37 64)(38 61)(39 62)(40 63)(53 57)(54 58)(55 59)(56 60)
(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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(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 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)
(1 53)(2 54)(3 55)(4 56)(5 52)(6 49)(7 50)(8 51)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 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 46 4 48)(5 58 7 60)(6 57 8 59)(9 19 11 17)(10 18 12 20)(13 23 15 21)(14 22 16 24)(25 36 27 34)(26 35 28 33)(29 40 31 38)(30 39 32 37)(41 51 43 49)(42 50 44 52)(53 63 55 61)(54 62 56 64)
(2 10)(4 12)(5 22)(6 51)(7 24)(8 49)(13 15)(14 44)(16 42)(17 36)(18 62)(19 34)(20 64)(21 38)(23 40)(26 56)(28 54)(29 31)(30 58)(32 60)(33 46)(35 48)(37 52)(39 50)(41 43)(45 61)(47 63)(57 59)
G:=sub<Sym(64)| (1,41)(2,42)(3,43)(4,44)(5,33)(6,34)(7,35)(8,36)(9,13)(10,14)(11,15)(12,16)(17,51)(18,52)(19,49)(20,50)(21,47)(22,48)(23,45)(24,46)(25,29)(26,30)(27,31)(28,32)(37,64)(38,61)(39,62)(40,63)(53,57)(54,58)(55,59)(56,60), (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), (1,9)(2,10)(3,11)(4,12)(5,39)(6,40)(7,37)(8,38)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,62)(34,63)(35,64)(36,61), (1,53)(2,54)(3,55)(4,56)(5,52)(6,49)(7,50)(8,51)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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)(2,46,4,48)(5,58,7,60)(6,57,8,59)(9,19,11,17)(10,18,12,20)(13,23,15,21)(14,22,16,24)(25,36,27,34)(26,35,28,33)(29,40,31,38)(30,39,32,37)(41,51,43,49)(42,50,44,52)(53,63,55,61)(54,62,56,64), (2,10)(4,12)(5,22)(6,51)(7,24)(8,49)(13,15)(14,44)(16,42)(17,36)(18,62)(19,34)(20,64)(21,38)(23,40)(26,56)(28,54)(29,31)(30,58)(32,60)(33,46)(35,48)(37,52)(39,50)(41,43)(45,61)(47,63)(57,59)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,33)(6,34)(7,35)(8,36)(9,13)(10,14)(11,15)(12,16)(17,51)(18,52)(19,49)(20,50)(21,47)(22,48)(23,45)(24,46)(25,29)(26,30)(27,31)(28,32)(37,64)(38,61)(39,62)(40,63)(53,57)(54,58)(55,59)(56,60), (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), (1,9)(2,10)(3,11)(4,12)(5,39)(6,40)(7,37)(8,38)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,62)(34,63)(35,64)(36,61), (1,53)(2,54)(3,55)(4,56)(5,52)(6,49)(7,50)(8,51)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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)(2,46,4,48)(5,58,7,60)(6,57,8,59)(9,19,11,17)(10,18,12,20)(13,23,15,21)(14,22,16,24)(25,36,27,34)(26,35,28,33)(29,40,31,38)(30,39,32,37)(41,51,43,49)(42,50,44,52)(53,63,55,61)(54,62,56,64), (2,10)(4,12)(5,22)(6,51)(7,24)(8,49)(13,15)(14,44)(16,42)(17,36)(18,62)(19,34)(20,64)(21,38)(23,40)(26,56)(28,54)(29,31)(30,58)(32,60)(33,46)(35,48)(37,52)(39,50)(41,43)(45,61)(47,63)(57,59) );
G=PermutationGroup([[(1,41),(2,42),(3,43),(4,44),(5,33),(6,34),(7,35),(8,36),(9,13),(10,14),(11,15),(12,16),(17,51),(18,52),(19,49),(20,50),(21,47),(22,48),(23,45),(24,46),(25,29),(26,30),(27,31),(28,32),(37,64),(38,61),(39,62),(40,63),(53,57),(54,58),(55,59),(56,60)], [(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)], [(1,9),(2,10),(3,11),(4,12),(5,39),(6,40),(7,37),(8,38),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,55),(26,56),(27,53),(28,54),(29,59),(30,60),(31,57),(32,58),(33,62),(34,63),(35,64),(36,61)], [(1,53),(2,54),(3,55),(4,56),(5,52),(6,49),(7,50),(8,51),(9,27),(10,28),(11,25),(12,26),(13,31),(14,32),(15,29),(16,30),(17,36),(18,33),(19,34),(20,35),(21,40),(22,37),(23,38),(24,39),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(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),(2,46,4,48),(5,58,7,60),(6,57,8,59),(9,19,11,17),(10,18,12,20),(13,23,15,21),(14,22,16,24),(25,36,27,34),(26,35,28,33),(29,40,31,38),(30,39,32,37),(41,51,43,49),(42,50,44,52),(53,63,55,61),(54,62,56,64)], [(2,10),(4,12),(5,22),(6,51),(7,24),(8,49),(13,15),(14,44),(16,42),(17,36),(18,62),(19,34),(20,64),(21,38),(23,40),(26,56),(28,54),(29,31),(30,58),(32,60),(33,46),(35,48),(37,52),(39,50),(41,43),(45,61),(47,63),(57,59)]])
Matrix representation of C23.714C24 ►in GL10(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 4 | 0 | 4 |
G:=sub<GL(10,GF(5))| [4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,4,1,0,0,0,0,0,0,0,0,3,0,1],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,2,2],[1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,4,3,0,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4] >;
C23.714C24 in GAP, Magma, Sage, TeX
C_2^3._{714}C_2^4 % in TeX
G:=Group("C2^3.714C2^4"); // GroupNames label
G:=SmallGroup(128,1546);
// by ID
G=gap.SmallGroup(128,1546);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,794,185]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=f^2=b,g*a*g=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a*b*c,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g=d*f=f*d,d*g=g*d>;
// generators/relations
Export