p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.502C24, C24.353C23, C22.2832+ 1+4, (C22×C4)⋊14D4, C23.186(C2×D4), C23.62(C4○D4), C23⋊2D4.14C2, C23.8Q8⋊79C2, C23.11D4⋊53C2, C23.23D4⋊63C2, C23.10D4⋊49C2, C2.20(C23⋊3D4), (C22×C4).123C23, (C2×C42).589C22, (C23×C4).132C22, C22.332(C22×D4), C24.3C22⋊61C2, (C22×D4).538C22, C23.65C23⋊98C2, C24.C22⋊100C2, C2.75(C22.19C24), C23.63C23⋊106C2, C2.68(C22.45C24), C2.C42.232C22, C2.45(C22.26C24), C2.31(C22.53C24), C2.78(C22.47C24), (C2×C4×D4)⋊50C2, (C4×C22⋊C4)⋊94C2, (C2×C4).1199(C2×D4), (C2×C4).161(C4○D4), (C2×C4⋊C4).342C22, C22.378(C2×C4○D4), (C2×C22.D4)⋊23C2, (C2×C22⋊C4).202C22, SmallGroup(128,1334)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.502C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=c, g2=db=bd, eae-1=ab=ba, ac=ca, faf=ad=da, ag=ga, bc=cb, geg-1=be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >
Subgroups: 596 in 291 conjugacy classes, 100 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C22.D4, C23×C4, C22×D4, C4×C22⋊C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.65C23, C24.3C22, C23⋊2D4, C23.10D4, C23.11D4, C2×C4×D4, C2×C22.D4, C23.502C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C22.19C24, C22.26C24, C23⋊3D4, C22.45C24, C22.47C24, C22.53C24, C23.502C24
►On 64 points
Generators in S64
(1 41)(2 56)(3 43)(4 54)(5 51)(6 35)(7 49)(8 33)(9 59)(10 46)(11 57)(12 48)(13 52)(14 36)(15 50)(16 34)(17 38)(18 29)(19 40)(20 31)(21 53)(22 44)(23 55)(24 42)(25 47)(26 58)(27 45)(28 60)(30 62)(32 64)(37 63)(39 61)
(1 23)(2 24)(3 21)(4 22)(5 16)(6 13)(7 14)(8 15)(9 27)(10 28)(11 25)(12 26)(17 64)(18 61)(19 62)(20 63)(29 39)(30 40)(31 37)(32 38)(33 50)(34 51)(35 52)(36 49)(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 25)(2 26)(3 27)(4 28)(5 38)(6 39)(7 40)(8 37)(9 21)(10 22)(11 23)(12 24)(13 29)(14 30)(15 31)(16 32)(17 51)(18 52)(19 49)(20 50)(33 63)(34 64)(35 61)(36 62)(41 47)(42 48)(43 45)(44 46)(53 59)(54 60)(55 57)(56 58)
(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 15)(6 14)(7 13)(8 16)(9 10)(11 12)(17 50)(18 49)(19 52)(20 51)(21 22)(23 24)(25 26)(27 28)(29 40)(30 39)(31 38)(32 37)(33 64)(34 63)(35 62)(36 61)(41 58)(42 57)(43 60)(44 59)(45 54)(46 53)(47 56)(48 55)
(1 29 11 6)(2 40 12 14)(3 31 9 8)(4 38 10 16)(5 22 32 28)(7 24 30 26)(13 23 39 25)(15 21 37 27)(17 46 34 54)(18 57 35 41)(19 48 36 56)(20 59 33 43)(42 62 58 49)(44 64 60 51)(45 50 53 63)(47 52 55 61)
G:=sub<Sym(64)| (1,41)(2,56)(3,43)(4,54)(5,51)(6,35)(7,49)(8,33)(9,59)(10,46)(11,57)(12,48)(13,52)(14,36)(15,50)(16,34)(17,38)(18,29)(19,40)(20,31)(21,53)(22,44)(23,55)(24,42)(25,47)(26,58)(27,45)(28,60)(30,62)(32,64)(37,63)(39,61), (1,23)(2,24)(3,21)(4,22)(5,16)(6,13)(7,14)(8,15)(9,27)(10,28)(11,25)(12,26)(17,64)(18,61)(19,62)(20,63)(29,39)(30,40)(31,37)(32,38)(33,50)(34,51)(35,52)(36,49)(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), (1,25)(2,26)(3,27)(4,28)(5,38)(6,39)(7,40)(8,37)(9,21)(10,22)(11,23)(12,24)(13,29)(14,30)(15,31)(16,32)(17,51)(18,52)(19,49)(20,50)(33,63)(34,64)(35,61)(36,62)(41,47)(42,48)(43,45)(44,46)(53,59)(54,60)(55,57)(56,58), (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)(5,15)(6,14)(7,13)(8,16)(9,10)(11,12)(17,50)(18,49)(19,52)(20,51)(21,22)(23,24)(25,26)(27,28)(29,40)(30,39)(31,38)(32,37)(33,64)(34,63)(35,62)(36,61)(41,58)(42,57)(43,60)(44,59)(45,54)(46,53)(47,56)(48,55), (1,29,11,6)(2,40,12,14)(3,31,9,8)(4,38,10,16)(5,22,32,28)(7,24,30,26)(13,23,39,25)(15,21,37,27)(17,46,34,54)(18,57,35,41)(19,48,36,56)(20,59,33,43)(42,62,58,49)(44,64,60,51)(45,50,53,63)(47,52,55,61)>;
G:=Group( (1,41)(2,56)(3,43)(4,54)(5,51)(6,35)(7,49)(8,33)(9,59)(10,46)(11,57)(12,48)(13,52)(14,36)(15,50)(16,34)(17,38)(18,29)(19,40)(20,31)(21,53)(22,44)(23,55)(24,42)(25,47)(26,58)(27,45)(28,60)(30,62)(32,64)(37,63)(39,61), (1,23)(2,24)(3,21)(4,22)(5,16)(6,13)(7,14)(8,15)(9,27)(10,28)(11,25)(12,26)(17,64)(18,61)(19,62)(20,63)(29,39)(30,40)(31,37)(32,38)(33,50)(34,51)(35,52)(36,49)(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), (1,25)(2,26)(3,27)(4,28)(5,38)(6,39)(7,40)(8,37)(9,21)(10,22)(11,23)(12,24)(13,29)(14,30)(15,31)(16,32)(17,51)(18,52)(19,49)(20,50)(33,63)(34,64)(35,61)(36,62)(41,47)(42,48)(43,45)(44,46)(53,59)(54,60)(55,57)(56,58), (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)(5,15)(6,14)(7,13)(8,16)(9,10)(11,12)(17,50)(18,49)(19,52)(20,51)(21,22)(23,24)(25,26)(27,28)(29,40)(30,39)(31,38)(32,37)(33,64)(34,63)(35,62)(36,61)(41,58)(42,57)(43,60)(44,59)(45,54)(46,53)(47,56)(48,55), (1,29,11,6)(2,40,12,14)(3,31,9,8)(4,38,10,16)(5,22,32,28)(7,24,30,26)(13,23,39,25)(15,21,37,27)(17,46,34,54)(18,57,35,41)(19,48,36,56)(20,59,33,43)(42,62,58,49)(44,64,60,51)(45,50,53,63)(47,52,55,61) );
G=PermutationGroup([[(1,41),(2,56),(3,43),(4,54),(5,51),(6,35),(7,49),(8,33),(9,59),(10,46),(11,57),(12,48),(13,52),(14,36),(15,50),(16,34),(17,38),(18,29),(19,40),(20,31),(21,53),(22,44),(23,55),(24,42),(25,47),(26,58),(27,45),(28,60),(30,62),(32,64),(37,63),(39,61)], [(1,23),(2,24),(3,21),(4,22),(5,16),(6,13),(7,14),(8,15),(9,27),(10,28),(11,25),(12,26),(17,64),(18,61),(19,62),(20,63),(29,39),(30,40),(31,37),(32,38),(33,50),(34,51),(35,52),(36,49),(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)], [(1,25),(2,26),(3,27),(4,28),(5,38),(6,39),(7,40),(8,37),(9,21),(10,22),(11,23),(12,24),(13,29),(14,30),(15,31),(16,32),(17,51),(18,52),(19,49),(20,50),(33,63),(34,64),(35,61),(36,62),(41,47),(42,48),(43,45),(44,46),(53,59),(54,60),(55,57),(56,58)], [(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),(5,15),(6,14),(7,13),(8,16),(9,10),(11,12),(17,50),(18,49),(19,52),(20,51),(21,22),(23,24),(25,26),(27,28),(29,40),(30,39),(31,38),(32,37),(33,64),(34,63),(35,62),(36,61),(41,58),(42,57),(43,60),(44,59),(45,54),(46,53),(47,56),(48,55)], [(1,29,11,6),(2,40,12,14),(3,31,9,8),(4,38,10,16),(5,22,32,28),(7,24,30,26),(13,23,39,25),(15,21,37,27),(17,46,34,54),(18,57,35,41),(19,48,36,56),(20,59,33,43),(42,62,58,49),(44,64,60,51),(45,50,53,63),(47,52,55,61)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X | 4Y |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C4○D4 | 2+ 1+4 |
| kernel | C23.502C24 | C4×C22⋊C4 | C23.8Q8 | C23.23D4 | C23.63C23 | C24.C22 | C23.65C23 | C24.3C22 | C23⋊2D4 | C23.10D4 | C23.11D4 | C2×C4×D4 | C2×C22.D4 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 12 | 4 | 2 |
Matrix representation of C23.502C24 ►in GL6(𝔽5)
| 2 | 2 | 0 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 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 |
| 0 | 4 | 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 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(6,GF(5))| [2,1,0,0,0,0,2,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,3,0,0,0,0,4,3],[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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,1,0,0,0,0,0,1],[4,2,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,2,4] >;
C23.502C24 in GAP, Magma, Sage, TeX
C_2^3._{502}C_2^4 % in TeX
G:=Group("C2^3.502C2^4"); // GroupNames label
G:=SmallGroup(128,1334);
// by ID
G=gap.SmallGroup(128,1334);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,675,304]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=c,g^2=d*b=b*d,e*a*e^-1=a*b=b*a,a*c=c*a,f*a*f=a*d=d*a,a*g=g*a,b*c=c*b,g*e*g^-1=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*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
// generators/relations