p-group, metabelian, nilpotent (class 2), monomial
Aliases: C82⋊2C2, C23.13C42, C42.746C23, C4⋊C8.21C4, C8⋊C8⋊22C2, C4.56(C8○D4), C22⋊C8.20C4, (C2×C4).17C42, C42.49(C2×C4), (C4×C8).307C22, C22.43(C2×C42), C2.9(C8○2M4(2)), (C2×C42).143C22, C42.12C4.44C2, (C2×C8).120(C2×C4), (C2×C4).583(C22×C4), (C22×C4).173(C2×C4), SmallGroup(128,186)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C82⋊2C2
G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a5b4, cbc=a4b >
Subgroups: 116 in 89 conjugacy classes, 64 normal (7 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C42, C2×C8, C22×C4, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C82, C8⋊C8, C42.12C4, C82⋊2C2
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C2×C42, C8○D4, C8○2M4(2), C82⋊2C2
►On 64 points
(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 42 34 12 57 29 55 17)(2 43 35 13 58 30 56 18)(3 44 36 14 59 31 49 19)(4 45 37 15 60 32 50 20)(5 46 38 16 61 25 51 21)(6 47 39 9 62 26 52 22)(7 48 40 10 63 27 53 23)(8 41 33 11 64 28 54 24)
(2 62)(4 64)(6 58)(8 60)(9 22)(10 14)(11 24)(12 16)(13 18)(15 20)(17 21)(19 23)(25 29)(26 47)(27 31)(28 41)(30 43)(32 45)(33 50)(35 52)(37 54)(39 56)(42 46)(44 48)
G:=sub<Sym(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), (1,42,34,12,57,29,55,17)(2,43,35,13,58,30,56,18)(3,44,36,14,59,31,49,19)(4,45,37,15,60,32,50,20)(5,46,38,16,61,25,51,21)(6,47,39,9,62,26,52,22)(7,48,40,10,63,27,53,23)(8,41,33,11,64,28,54,24), (2,62)(4,64)(6,58)(8,60)(9,22)(10,14)(11,24)(12,16)(13,18)(15,20)(17,21)(19,23)(25,29)(26,47)(27,31)(28,41)(30,43)(32,45)(33,50)(35,52)(37,54)(39,56)(42,46)(44,48)>;
G:=Group( (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), (1,42,34,12,57,29,55,17)(2,43,35,13,58,30,56,18)(3,44,36,14,59,31,49,19)(4,45,37,15,60,32,50,20)(5,46,38,16,61,25,51,21)(6,47,39,9,62,26,52,22)(7,48,40,10,63,27,53,23)(8,41,33,11,64,28,54,24), (2,62)(4,64)(6,58)(8,60)(9,22)(10,14)(11,24)(12,16)(13,18)(15,20)(17,21)(19,23)(25,29)(26,47)(27,31)(28,41)(30,43)(32,45)(33,50)(35,52)(37,54)(39,56)(42,46)(44,48) );
G=PermutationGroup([[(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)], [(1,42,34,12,57,29,55,17),(2,43,35,13,58,30,56,18),(3,44,36,14,59,31,49,19),(4,45,37,15,60,32,50,20),(5,46,38,16,61,25,51,21),(6,47,39,9,62,26,52,22),(7,48,40,10,63,27,53,23),(8,41,33,11,64,28,54,24)], [(2,62),(4,64),(6,58),(8,60),(9,22),(10,14),(11,24),(12,16),(13,18),(15,20),(17,21),(19,23),(25,29),(26,47),(27,31),(28,41),(30,43),(32,45),(33,50),(35,52),(37,54),(39,56),(42,46),(44,48)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4L | 4M | 4N | 4O | 8A | ··· | 8X | 8Y | ··· | 8AJ |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8○D4 |
| kernel | C82⋊2C2 | C82 | C8⋊C8 | C42.12C4 | C22⋊C8 | C4⋊C8 | C4 |
| # reps | 1 | 1 | 3 | 3 | 12 | 12 | 24 |
Matrix representation of C82⋊2C2 ►in GL4(𝔽17) generated by
| 1 | 16 | 0 | 0 |
| 2 | 16 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 15 | 1 |
| 0 | 0 | 9 | 2 |
| 1 | 16 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 16 |
G:=sub<GL(4,GF(17))| [1,2,0,0,16,16,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,8,0,0,0,0,15,9,0,0,1,2],[1,0,0,0,16,16,0,0,0,0,1,4,0,0,0,16] >;
C82⋊2C2 in GAP, Magma, Sage, TeX
C_8^2\rtimes_2C_2
% in TeX
G:=Group("C8^2:2C2"); // GroupNames label
G:=SmallGroup(128,186);
// by ID
G=gap.SmallGroup(128,186);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,120,2102,387,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a^5*b^4,c*b*c=a^4*b>;
// generators/relations