direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C2×C8○D8, C42.687C23, M4(2).30C23, (C2×C8)○D8, C8○(C2×D8), C8○(C2×Q16), (C2×C8)○Q16, C8○(C8○D8), C4○(C8○D8), (C2×C8)○SD16, C8○(C2×SD16), C8○2(C4○D8), C4○D8⋊10C4, (C2×D8)⋊21C4, D8⋊17(C2×C4), C4.48(C4×D4), (C4×C8)⋊75C22, Q16⋊16(C2×C4), (C2×Q16)⋊21C4, (C2×C8).405D4, C8.145(C2×D4), C4≀C2⋊20C22, C22.6(C4×D4), (C2×SD16)⋊18C4, SD16⋊13(C2×C4), C8○D4⋊17C22, C8.44(C22×C4), C4.30(C23×C4), (C2×C4).210C24, (C2×C8).584C23, C4○D8.32C22, C4○D4.22C23, D4.12(C22×C4), C4.201(C22×D4), Q8.12(C22×C4), C8.C4⋊22C22, C23.256(C4○D4), (C22×C8).556C22, (C2×C42).1115C22, (C22×C4).1517C23, (C2×M4(2)).357C22, (C2×C4×C8)⋊32C2, C8○(C2×C4○D8), (C2×C8)○(C2×D8), C8○2(C2×C4≀C2), (C2×C8)○2C4≀C2, (C2×C8)○(C2×Q16), C2.70(C2×C4×D4), (C2×C8)○(C4○D8), (C2×C4≀C2)⋊33C2, (C2×C8)○(C2×SD16), C8○2(C2×C8.C4), (C2×C8○D4)⋊25C2, (C2×C8).187(C2×C4), C4○D4.21(C2×C4), (C2×C4○D8).22C2, (C2×C8.C4)⋊29C2, C22.1(C2×C4○D4), (C2×C8)○2(C8.C4), (C2×D4).177(C2×C4), (C2×C4).1577(C2×D4), (C2×Q8).160(C2×C4), (C2×C4).693(C4○D4), (C2×C4).472(C22×C4), (C2×C4○D4).293C22, (C2×C8)○(C2×C4≀C2), (C2×C8)○(C2×C4○D8), (C2×C8)○(C2×C8.C4), SmallGroup(128,1685)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C8○D8
G = < a,b,c,d | a2=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c3 >
Subgroups: 348 in 236 conjugacy classes, 140 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C2×C8, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4×C8, C4≀C2, C8.C4, C2×C42, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C4×C8, C2×C4≀C2, C2×C8.C4, C8○D8, C2×C8○D4, C2×C4○D8, C2×C8○D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, C8○D8, C2×C4×D4, C2×C8○D8
►On 32 points
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)
(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)
(1 20 3 22 5 24 7 18)(2 21 4 23 6 17 8 19)(9 31 15 29 13 27 11 25)(10 32 16 30 14 28 12 26)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
G:=sub<Sym(32)| (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (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), (1,20,3,22,5,24,7,18)(2,21,4,23,6,17,8,19)(9,31,15,29,13,27,11,25)(10,32,16,30,14,28,12,26), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)>;
G:=Group( (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27), (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), (1,20,3,22,5,24,7,18)(2,21,4,23,6,17,8,19)(9,31,15,29,13,27,11,25)(10,32,16,30,14,28,12,26), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20) );
G=PermutationGroup([[(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27)], [(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)], [(1,20,3,22,5,24,7,18),(2,21,4,23,6,17,8,19),(9,31,15,29,13,27,11,25),(10,32,16,30,14,28,12,26)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8H | 8I | ··· | 8T | 8U | ··· | 8AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | C4○D4 | C4○D4 | C8○D8 |
| kernel | C2×C8○D8 | C2×C4×C8 | C2×C4≀C2 | C2×C8.C4 | C8○D8 | C2×C8○D4 | C2×C4○D8 | C2×D8 | C2×SD16 | C2×Q16 | C4○D8 | C2×C8 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 2 | 4 | 2 | 8 | 4 | 2 | 2 | 16 |
Matrix representation of C2×C8○D8 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 4 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 2 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[4,0,0,0,9,0,0,0,9],[1,0,0,0,9,0,0,0,2],[1,0,0,0,0,1,0,1,0] >;
C2×C8○D8 in GAP, Magma, Sage, TeX
C_2\times C_8\circ D_8
% in TeX
G:=Group("C2xC8oD8"); // GroupNames label
G:=SmallGroup(128,1685);
// by ID
G=gap.SmallGroup(128,1685);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2804,1411,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^3>;
// generators/relations