direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8⋊5D4, C42.356D4, C42.710C23, (C2×C8)⋊32D4, C8⋊11(C2×D4), C4⋊1(C2×SD16), (C4×C8)⋊77C22, (C2×C4)⋊10SD16, C4⋊Q8⋊65C22, C4.1(C22×D4), C4.11(C4⋊1D4), (C2×C4).341C24, (C2×C8).592C23, (C22×C4).611D4, C23.876(C2×D4), (C2×Q8).96C23, (C22×SD16)⋊26C2, (C2×SD16)⋊77C22, (C2×D4).108C23, C22.88(C2×SD16), C2.18(C22×SD16), C4⋊1D4.150C22, C22.47(C4⋊1D4), (C22×C8).566C22, C22.601(C22×D4), (C2×C42).1125C22, (C22×C4).1556C23, (C22×D4).371C22, (C22×Q8).304C22, (C2×C4×C8)⋊35C2, (C2×C4⋊Q8)⋊37C2, (C2×C4).851(C2×D4), C2.20(C2×C4⋊1D4), (C2×C4⋊1D4).25C2, SmallGroup(128,1875)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8⋊5D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b3, dcd=c-1 >
Subgroups: 724 in 328 conjugacy classes, 132 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4×C8, C2×C42, C2×C4⋊C4, C4⋊1D4, C4⋊1D4, C4⋊Q8, C4⋊Q8, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×D4, C22×Q8, C2×C4×C8, C8⋊5D4, C2×C4⋊1D4, C2×C4⋊Q8, C22×SD16, C2×C8⋊5D4
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C24, C4⋊1D4, C2×SD16, C22×D4, C8⋊5D4, C2×C4⋊1D4, C22×SD16, C2×C8⋊5D4
►On 64 points
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 39)(10 40)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)
(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 26 63 13)(2 27 64 14)(3 28 57 15)(4 29 58 16)(5 30 59 9)(6 31 60 10)(7 32 61 11)(8 25 62 12)(17 34 56 42)(18 35 49 43)(19 36 50 44)(20 37 51 45)(21 38 52 46)(22 39 53 47)(23 40 54 48)(24 33 55 41)
(1 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)(17 48)(18 43)(19 46)(20 41)(21 44)(22 47)(23 42)(24 45)(25 60)(26 63)(27 58)(28 61)(29 64)(30 59)(31 62)(32 57)(33 51)(34 54)(35 49)(36 52)(37 55)(38 50)(39 53)(40 56)
G:=sub<Sym(64)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (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), (1,26,63,13)(2,27,64,14)(3,28,57,15)(4,29,58,16)(5,30,59,9)(6,31,60,10)(7,32,61,11)(8,25,62,12)(17,34,56,42)(18,35,49,43)(19,36,50,44)(20,37,51,45)(21,38,52,46)(22,39,53,47)(23,40,54,48)(24,33,55,41), (1,13)(2,16)(3,11)(4,14)(5,9)(6,12)(7,15)(8,10)(17,48)(18,43)(19,46)(20,41)(21,44)(22,47)(23,42)(24,45)(25,60)(26,63)(27,58)(28,61)(29,64)(30,59)(31,62)(32,57)(33,51)(34,54)(35,49)(36,52)(37,55)(38,50)(39,53)(40,56)>;
G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,39)(10,40)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41), (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), (1,26,63,13)(2,27,64,14)(3,28,57,15)(4,29,58,16)(5,30,59,9)(6,31,60,10)(7,32,61,11)(8,25,62,12)(17,34,56,42)(18,35,49,43)(19,36,50,44)(20,37,51,45)(21,38,52,46)(22,39,53,47)(23,40,54,48)(24,33,55,41), (1,13)(2,16)(3,11)(4,14)(5,9)(6,12)(7,15)(8,10)(17,48)(18,43)(19,46)(20,41)(21,44)(22,47)(23,42)(24,45)(25,60)(26,63)(27,58)(28,61)(29,64)(30,59)(31,62)(32,57)(33,51)(34,54)(35,49)(36,52)(37,55)(38,50)(39,53)(40,56) );
G=PermutationGroup([[(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,39),(10,40),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,62),(18,63),(19,64),(20,57),(21,58),(22,59),(23,60),(24,61),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41)], [(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)], [(1,26,63,13),(2,27,64,14),(3,28,57,15),(4,29,58,16),(5,30,59,9),(6,31,60,10),(7,32,61,11),(8,25,62,12),(17,34,56,42),(18,35,49,43),(19,36,50,44),(20,37,51,45),(21,38,52,46),(22,39,53,47),(23,40,54,48),(24,33,55,41)], [(1,13),(2,16),(3,11),(4,14),(5,9),(6,12),(7,15),(8,10),(17,48),(18,43),(19,46),(20,41),(21,44),(22,47),(23,42),(24,45),(25,60),(26,63),(27,58),(28,61),(29,64),(30,59),(31,62),(32,57),(33,51),(34,54),(35,49),(36,52),(37,55),(38,50),(39,53),(40,56)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | SD16 |
| kernel | C2×C8⋊5D4 | C2×C4×C8 | C8⋊5D4 | C2×C4⋊1D4 | C2×C4⋊Q8 | C22×SD16 | C42 | C2×C8 | C22×C4 | C2×C4 |
| # reps | 1 | 1 | 8 | 1 | 1 | 4 | 2 | 8 | 2 | 16 |
Matrix representation of C2×C8⋊5D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
C2×C8⋊5D4 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes_5D_4
% in TeX
G:=Group("C2xC8:5D4"); // GroupNames label
G:=SmallGroup(128,1875);
// by ID
G=gap.SmallGroup(128,1875);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,184,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations