p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).168D4, (C2×D4).14Q8, C42⋊8C4⋊9C2, C2.16(C8⋊3D4), C23.939(C2×D4), (C22×C4).164D4, C2.14(D4.Q8), C4.39(C22⋊Q8), C2.18(C8.12D4), C22.129(C4○D8), C22.78(C4⋊1D4), (C2×C42).390C22, (C22×C8).330C22, (C22×D4).95C22, C2.5(C23.4Q8), C22.158(C8⋊C22), (C22×C4).1473C23, C22.7C42⋊22C2, C4.33(C22.D4), C22.116(C22⋊Q8), C2.14(C23.19D4), C24.3C22.21C2, C22.126(C22.D4), (C2×C4.Q8)⋊25C2, (C2×C2.D8)⋊13C2, (C2×C4).748(C2×D4), (C2×C4).290(C2×Q8), (C2×D4⋊C4).22C2, (C2×C4).779(C4○D4), (C2×C4⋊C4).162C22, SmallGroup(128,824)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).168D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab5, dbd-1=b3, dcd-1=b4c3 >
Subgroups: 336 in 137 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, D4⋊C4, C4.Q8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C22.7C42, C42⋊8C4, C24.3C22, C2×D4⋊C4, C2×C4.Q8, C2×C2.D8, (C2×C8).168D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C22.D4, C4⋊1D4, C4○D8, C8⋊C22, C23.4Q8, D4.Q8, C23.19D4, C8.12D4, C8⋊3D4, (C2×C8).168D4
►On 64 points
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)
(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 21 40 31 5 17 36 27)(2 10 33 47 6 14 37 43)(3 23 34 25 7 19 38 29)(4 12 35 41 8 16 39 45)(9 56 46 63 13 52 42 59)(11 50 48 57 15 54 44 61)(18 53 28 60 22 49 32 64)(20 55 30 62 24 51 26 58)
(1 28 63 47)(2 31 64 42)(3 26 57 45)(4 29 58 48)(5 32 59 43)(6 27 60 46)(7 30 61 41)(8 25 62 44)(9 37 17 49)(10 40 18 52)(11 35 19 55)(12 38 20 50)(13 33 21 53)(14 36 22 56)(15 39 23 51)(16 34 24 54)
G:=sub<Sym(64)| (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52), (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), (1,21,40,31,5,17,36,27)(2,10,33,47,6,14,37,43)(3,23,34,25,7,19,38,29)(4,12,35,41,8,16,39,45)(9,56,46,63,13,52,42,59)(11,50,48,57,15,54,44,61)(18,53,28,60,22,49,32,64)(20,55,30,62,24,51,26,58), (1,28,63,47)(2,31,64,42)(3,26,57,45)(4,29,58,48)(5,32,59,43)(6,27,60,46)(7,30,61,41)(8,25,62,44)(9,37,17,49)(10,40,18,52)(11,35,19,55)(12,38,20,50)(13,33,21,53)(14,36,22,56)(15,39,23,51)(16,34,24,54)>;
G:=Group( (1,63)(2,64)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52), (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), (1,21,40,31,5,17,36,27)(2,10,33,47,6,14,37,43)(3,23,34,25,7,19,38,29)(4,12,35,41,8,16,39,45)(9,56,46,63,13,52,42,59)(11,50,48,57,15,54,44,61)(18,53,28,60,22,49,32,64)(20,55,30,62,24,51,26,58), (1,28,63,47)(2,31,64,42)(3,26,57,45)(4,29,58,48)(5,32,59,43)(6,27,60,46)(7,30,61,41)(8,25,62,44)(9,37,17,49)(10,40,18,52)(11,35,19,55)(12,38,20,50)(13,33,21,53)(14,36,22,56)(15,39,23,51)(16,34,24,54) );
G=PermutationGroup([[(1,63),(2,64),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(25,44),(26,45),(27,46),(28,47),(29,48),(30,41),(31,42),(32,43),(33,53),(34,54),(35,55),(36,56),(37,49),(38,50),(39,51),(40,52)], [(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)], [(1,21,40,31,5,17,36,27),(2,10,33,47,6,14,37,43),(3,23,34,25,7,19,38,29),(4,12,35,41,8,16,39,45),(9,56,46,63,13,52,42,59),(11,50,48,57,15,54,44,61),(18,53,28,60,22,49,32,64),(20,55,30,62,24,51,26,58)], [(1,28,63,47),(2,31,64,42),(3,26,57,45),(4,29,58,48),(5,32,59,43),(6,27,60,46),(7,30,61,41),(8,25,62,44),(9,37,17,49),(10,40,18,52),(11,35,19,55),(12,38,20,50),(13,33,21,53),(14,36,22,56),(15,39,23,51),(16,34,24,54)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | C4○D8 | C8⋊C22 |
| kernel | (C2×C8).168D4 | C22.7C42 | C42⋊8C4 | C24.3C22 | C2×D4⋊C4 | C2×C4.Q8 | C2×C2.D8 | C2×C8 | C22×C4 | C2×D4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 6 | 8 | 2 |
Matrix representation of (C2×C8).168D4 ►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 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 14 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 13 | 0 | 0 |
| 0 | 0 | 3 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 |
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],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,2,14,0,0,0,0,1,15,0,0,0,0,0,0,5,5,0,0,0,0,12,5],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,9,3,0,0,0,0,13,8,0,0,0,0,0,0,14,3,0,0,0,0,14,14],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0] >;
(C2×C8).168D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)._{168}D_4 % in TeX
G:=Group("(C2xC8).168D4"); // GroupNames label
G:=SmallGroup(128,824);
// by ID
G=gap.SmallGroup(128,824);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,176,422,387,1018,521,248,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^5,d*b*d^-1=b^3,d*c*d^-1=b^4*c^3>;
// generators/relations