p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8)⋊3D4, (C2×D4).77D4, (C2×Q8).72D4, C22≀C2.2C4, Q8○M4(2)⋊7C2, C24.58(C2×C4), C22.40(C4×D4), (C22×C4).66D4, C4.127C22≀C2, C4.132(C4⋊D4), C24.4C4⋊26C2, C23.69(C22×C4), C22.D4.2C4, M4(2)⋊4C4⋊15C2, C23.12(C22⋊C4), C22.19C24.6C2, (C22×C4).687C23, (C23×C4).258C22, C42⋊C2.24C22, C4.109(C22.D4), C2.30(C23.23D4), (C2×M4(2)).184C22, M4(2).8C22⋊10C2, (C2×D4).80(C2×C4), C22⋊C4.5(C2×C4), (C2×C4).1330(C2×D4), (C2×C4).321(C4○D4), (C2×C4).16(C22⋊C4), (C22×C4).120(C2×C4), (C2×C4○D4).20C22, C22.46(C2×C22⋊C4), SmallGroup(128,623)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8)⋊D4
G = < a,b,c,d | a2=b8=c4=d2=1, cbc-1=dbd=ab=ba, cac-1=ab4, ad=da, dcd=c-1 >
Subgroups: 340 in 171 conjugacy classes, 54 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, C4.D4, C4.10D4, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C2×M4(2), C2×M4(2), C8○D4, C23×C4, C2×C4○D4, M4(2)⋊4C4, C24.4C4, M4(2).8C22, C22.19C24, Q8○M4(2), (C2×C8)⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, (C2×C8)⋊D4
►On 16 points - transitive group 16T209
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 7 5 3)(2 14)(4 16)(6 10)(8 12)(9 11 13 15)
(1 15)(2 6)(3 9)(4 8)(5 11)(7 13)(10 14)(12 16)
G:=sub<Sym(16)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,7,5,3)(2,14)(4,16)(6,10)(8,12)(9,11,13,15), (1,15)(2,6)(3,9)(4,8)(5,11)(7,13)(10,14)(12,16)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,7,5,3)(2,14)(4,16)(6,10)(8,12)(9,11,13,15), (1,15)(2,6)(3,9)(4,8)(5,11)(7,13)(10,14)(12,16) );
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,7,5,3),(2,14),(4,16),(6,10),(8,12),(9,11,13,15)], [(1,15),(2,6),(3,9),(4,8),(5,11),(7,13),(10,14),(12,16)]])
G:=TransitiveGroup(16,209);
►On 16 points - transitive group 16T279
(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 11 5 15)(2 12)(3 13 7 9)(4 14)(6 16)(8 10)
(9 13)(11 15)
G:=sub<Sym(16)| (9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11,5,15)(2,12)(3,13,7,9)(4,14)(6,16)(8,10), (9,13)(11,15)>;
G:=Group( (9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11,5,15)(2,12)(3,13,7,9)(4,14)(6,16)(8,10), (9,13)(11,15) );
G=PermutationGroup([[(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,11,5,15),(2,12),(3,13,7,9),(4,14),(6,16),(8,10)], [(9,13),(11,15)]])
G:=TransitiveGroup(16,279);
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D4 | D4 | C4○D4 | (C2×C8)⋊D4 |
| kernel | (C2×C8)⋊D4 | M4(2)⋊4C4 | C24.4C4 | M4(2).8C22 | C22.19C24 | Q8○M4(2) | C22≀C2 | C22.D4 | C2×C8 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 4 | 4 |
Matrix representation of (C2×C8)⋊D4 ►in GL4(𝔽5) generated by
| 0 | 4 | 3 | 3 |
| 0 | 3 | 1 | 0 |
| 0 | 2 | 2 | 0 |
| 2 | 4 | 0 | 0 |
| 0 | 3 | 4 | 2 |
| 1 | 0 | 0 | 4 |
| 0 | 0 | 3 | 1 |
| 0 | 0 | 4 | 2 |
| 0 | 1 | 2 | 0 |
| 2 | 3 | 4 | 0 |
| 4 | 1 | 1 | 4 |
| 0 | 1 | 2 | 1 |
| 0 | 0 | 1 | 3 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 2 | 0 | 3 | 0 |
G:=sub<GL(4,GF(5))| [0,0,0,2,4,3,2,4,3,1,2,0,3,0,0,0],[0,1,0,0,3,0,0,0,4,0,3,4,2,4,1,2],[0,2,4,0,1,3,1,1,2,4,1,2,0,0,4,1],[0,0,0,2,0,1,0,0,1,0,1,3,3,0,0,0] >;
(C2×C8)⋊D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)\rtimes D_4
% in TeX
G:=Group("(C2xC8):D4"); // GroupNames label
G:=SmallGroup(128,623);
// by ID
G=gap.SmallGroup(128,623);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,521,2804,1411,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=d^2=1,c*b*c^-1=d*b*d=a*b=b*a,c*a*c^-1=a*b^4,a*d=d*a,d*c*d=c^-1>;
// generators/relations