p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C24.459C23, C23.713C24, C22.4862+ 1+4, C22.3712- 1+4, (C22×C4)⋊40D4, C23⋊2D4⋊46C2, C23.223(C2×D4), C23.Q8⋊91C2, C23.4Q8⋊64C2, C2.60(C23⋊3D4), (C23×C4).498C22, (C22×C4).224C23, C23.7Q8⋊113C2, C22.445(C22×D4), C23.10D4⋊107C2, (C22×D4).291C22, C2.67(C22.29C24), C23.83C23⋊130C2, C2.43(C22.54C24), C2.C42.416C22, C2.48(C22.56C24), C2.56(C22.31C24), (C2×C4⋊D4)⋊39C2, (C2×C4).430(C2×D4), (C2×C4⋊C4).522C22, (C2×C22.D4)⋊44C2, (C2×C22⋊C4).332C22, SmallGroup(128,1545)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.459C23
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=a, ab=ba, ac=ca, ede=ad=da, ae=ea, gfg=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, geg=abe >
Subgroups: 692 in 289 conjugacy classes, 92 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C2.C42, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4⋊D4, C22.D4, C23×C4, C22×D4, C22×D4, C23.7Q8, C23⋊2D4, C23⋊2D4, C23.10D4, C23.10D4, C23.Q8, C23.4Q8, C23.83C23, C2×C4⋊D4, C2×C22.D4, C24.459C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, 2- 1+4, C23⋊3D4, C22.29C24, C22.31C24, C22.54C24, C22.56C24, C24.459C23
Character table of C24.459C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 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 | 0 | 0 | 2 | 2 | -2 | -2 | 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 | 0 | 0 | 2 | -2 | 2 | -2 | 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 | 0 | 0 | -2 | 2 | -2 | 2 | 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 | 0 | 0 | -2 | -2 | 2 | 2 | 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 | orthogonal lifted from 2+ 1+4 |
| 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 | orthogonal lifted from 2+ 1+4 |
| 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 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 42)(2 43)(3 44)(4 41)(5 21)(6 22)(7 23)(8 24)(9 63)(10 64)(11 61)(12 62)(13 59)(14 60)(15 57)(16 58)(17 34)(18 35)(19 36)(20 33)(25 52)(26 49)(27 50)(28 51)(29 46)(30 47)(31 48)(32 45)(37 54)(38 55)(39 56)(40 53)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)(17 40)(18 37)(19 38)(20 39)(21 62)(22 63)(23 64)(24 61)(25 48)(26 45)(27 46)(28 47)(29 50)(30 51)(31 52)(32 49)(33 56)(34 53)(35 54)(36 55)(41 60)(42 57)(43 58)(44 59)
(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 2)(3 4)(5 61)(6 64)(7 63)(8 62)(9 23)(10 22)(11 21)(12 24)(13 14)(15 16)(17 39)(18 38)(19 37)(20 40)(25 49)(26 52)(27 51)(28 50)(29 47)(30 46)(31 45)(32 48)(33 53)(34 56)(35 55)(36 54)(41 44)(42 43)(57 58)(59 60)
(1 6)(2 23)(3 8)(4 21)(5 41)(7 43)(9 15)(10 58)(11 13)(12 60)(14 62)(16 64)(17 49)(18 27)(19 51)(20 25)(22 42)(24 44)(26 34)(28 36)(29 54)(30 38)(31 56)(32 40)(33 52)(35 50)(37 46)(39 48)(45 53)(47 55)(57 63)(59 61)
(1 51)(2 52)(3 49)(4 50)(5 20)(6 17)(7 18)(8 19)(9 40)(10 37)(11 38)(12 39)(13 32)(14 29)(15 30)(16 31)(21 33)(22 34)(23 35)(24 36)(25 43)(26 44)(27 41)(28 42)(45 59)(46 60)(47 57)(48 58)(53 63)(54 64)(55 61)(56 62)
G:=sub<Sym(64)| (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), (1,42)(2,43)(3,44)(4,41)(5,21)(6,22)(7,23)(8,24)(9,63)(10,64)(11,61)(12,62)(13,59)(14,60)(15,57)(16,58)(17,34)(18,35)(19,36)(20,33)(25,52)(26,49)(27,50)(28,51)(29,46)(30,47)(31,48)(32,45)(37,54)(38,55)(39,56)(40,53), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11)(17,40)(18,37)(19,38)(20,39)(21,62)(22,63)(23,64)(24,61)(25,48)(26,45)(27,46)(28,47)(29,50)(30,51)(31,52)(32,49)(33,56)(34,53)(35,54)(36,55)(41,60)(42,57)(43,58)(44,59), (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), (1,2)(3,4)(5,61)(6,64)(7,63)(8,62)(9,23)(10,22)(11,21)(12,24)(13,14)(15,16)(17,39)(18,38)(19,37)(20,40)(25,49)(26,52)(27,51)(28,50)(29,47)(30,46)(31,45)(32,48)(33,53)(34,56)(35,55)(36,54)(41,44)(42,43)(57,58)(59,60), (1,6)(2,23)(3,8)(4,21)(5,41)(7,43)(9,15)(10,58)(11,13)(12,60)(14,62)(16,64)(17,49)(18,27)(19,51)(20,25)(22,42)(24,44)(26,34)(28,36)(29,54)(30,38)(31,56)(32,40)(33,52)(35,50)(37,46)(39,48)(45,53)(47,55)(57,63)(59,61), (1,51)(2,52)(3,49)(4,50)(5,20)(6,17)(7,18)(8,19)(9,40)(10,37)(11,38)(12,39)(13,32)(14,29)(15,30)(16,31)(21,33)(22,34)(23,35)(24,36)(25,43)(26,44)(27,41)(28,42)(45,59)(46,60)(47,57)(48,58)(53,63)(54,64)(55,61)(56,62)>;
G:=Group( (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), (1,42)(2,43)(3,44)(4,41)(5,21)(6,22)(7,23)(8,24)(9,63)(10,64)(11,61)(12,62)(13,59)(14,60)(15,57)(16,58)(17,34)(18,35)(19,36)(20,33)(25,52)(26,49)(27,50)(28,51)(29,46)(30,47)(31,48)(32,45)(37,54)(38,55)(39,56)(40,53), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11)(17,40)(18,37)(19,38)(20,39)(21,62)(22,63)(23,64)(24,61)(25,48)(26,45)(27,46)(28,47)(29,50)(30,51)(31,52)(32,49)(33,56)(34,53)(35,54)(36,55)(41,60)(42,57)(43,58)(44,59), (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), (1,2)(3,4)(5,61)(6,64)(7,63)(8,62)(9,23)(10,22)(11,21)(12,24)(13,14)(15,16)(17,39)(18,38)(19,37)(20,40)(25,49)(26,52)(27,51)(28,50)(29,47)(30,46)(31,45)(32,48)(33,53)(34,56)(35,55)(36,54)(41,44)(42,43)(57,58)(59,60), (1,6)(2,23)(3,8)(4,21)(5,41)(7,43)(9,15)(10,58)(11,13)(12,60)(14,62)(16,64)(17,49)(18,27)(19,51)(20,25)(22,42)(24,44)(26,34)(28,36)(29,54)(30,38)(31,56)(32,40)(33,52)(35,50)(37,46)(39,48)(45,53)(47,55)(57,63)(59,61), (1,51)(2,52)(3,49)(4,50)(5,20)(6,17)(7,18)(8,19)(9,40)(10,37)(11,38)(12,39)(13,32)(14,29)(15,30)(16,31)(21,33)(22,34)(23,35)(24,36)(25,43)(26,44)(27,41)(28,42)(45,59)(46,60)(47,57)(48,58)(53,63)(54,64)(55,61)(56,62) );
G=PermutationGroup([[(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)], [(1,42),(2,43),(3,44),(4,41),(5,21),(6,22),(7,23),(8,24),(9,63),(10,64),(11,61),(12,62),(13,59),(14,60),(15,57),(16,58),(17,34),(18,35),(19,36),(20,33),(25,52),(26,49),(27,50),(28,51),(29,46),(30,47),(31,48),(32,45),(37,54),(38,55),(39,56),(40,53)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11),(17,40),(18,37),(19,38),(20,39),(21,62),(22,63),(23,64),(24,61),(25,48),(26,45),(27,46),(28,47),(29,50),(30,51),(31,52),(32,49),(33,56),(34,53),(35,54),(36,55),(41,60),(42,57),(43,58),(44,59)], [(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)], [(1,2),(3,4),(5,61),(6,64),(7,63),(8,62),(9,23),(10,22),(11,21),(12,24),(13,14),(15,16),(17,39),(18,38),(19,37),(20,40),(25,49),(26,52),(27,51),(28,50),(29,47),(30,46),(31,45),(32,48),(33,53),(34,56),(35,55),(36,54),(41,44),(42,43),(57,58),(59,60)], [(1,6),(2,23),(3,8),(4,21),(5,41),(7,43),(9,15),(10,58),(11,13),(12,60),(14,62),(16,64),(17,49),(18,27),(19,51),(20,25),(22,42),(24,44),(26,34),(28,36),(29,54),(30,38),(31,56),(32,40),(33,52),(35,50),(37,46),(39,48),(45,53),(47,55),(57,63),(59,61)], [(1,51),(2,52),(3,49),(4,50),(5,20),(6,17),(7,18),(8,19),(9,40),(10,37),(11,38),(12,39),(13,32),(14,29),(15,30),(16,31),(21,33),(22,34),(23,35),(24,36),(25,43),(26,44),(27,41),(28,42),(45,59),(46,60),(47,57),(48,58),(53,63),(54,64),(55,61),(56,62)]])
Matrix representation of C24.459C23 ►in GL10(ℤ)
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 0 | 0 | 0 | -1 | 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 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 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 | 0 | 1 | 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 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | 0 | 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 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| -1 | 0 | 0 | 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 | 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 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -1 |
G:=sub<GL(10,Integers())| [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,-1,0,0,0,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,-2,-1,0],[-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-2,0,0,-1,0,0,0,0,0,0,0,-2,-1,0],[0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1] >;
C24.459C23 in GAP, Magma, Sage, TeX
C_2^4._{459}C_2^3 % in TeX
G:=Group("C2^4.459C2^3"); // GroupNames label
G:=SmallGroup(128,1545);
// by ID
G=gap.SmallGroup(128,1545);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,794,185]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=g^2=1,d^2=a,a*b=b*a,a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,g*f*g=a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,g*e*g=a*b*e>;
// generators/relations
Export