direct product, p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C2×D4⋊4D4, C42⋊6C23, C24.42D4, M4(2)⋊1C23, 2+ 1+4⋊5C22, C4○D4⋊5D4, D4⋊9(C2×D4), Q8⋊9(C2×D4), (C2×D4)⋊51D4, (C2×Q8)⋊38D4, C4≀C2⋊7C22, (C2×D4)⋊3C23, (C2×C4).6C24, C4.43C22≀C2, C8⋊C22⋊4C22, C4○D4.1C23, C4.51(C22×D4), C23.19(C2×D4), C4⋊1D4⋊30C22, (C2×C42)⋊36C22, C4.D4⋊6C22, (C22×D4)⋊19C22, (C2×M4(2))⋊7C22, (C2×2+ 1+4)⋊3C2, C22.30(C22×D4), C22.118C22≀C2, (C22×C4).965C23, (C2×C4≀C2)⋊4C2, (C2×C4⋊1D4)⋊14C2, (C2×C8⋊C22)⋊11C2, (C2×C4.D4)⋊7C2, C2.51(C2×C22≀C2), (C2×C4).1095(C2×D4), (C2×C4○D4).104C22, SmallGroup(128,1746)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×D4⋊4D4
G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b-1c, ece=bc, ede=d-1 >
Subgroups: 980 in 422 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C42, C42, C2×C8, M4(2), M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C24, C4.D4, C4≀C2, C2×C42, C4⋊1D4, C4⋊1D4, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C8⋊C22, C22×D4, C22×D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, C2×C4.D4, C2×C4≀C2, D4⋊4D4, C2×C4⋊1D4, C2×C8⋊C22, C2×2+ 1+4, C2×D4⋊4D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, D4⋊4D4, C2×C22≀C2, C2×D4⋊4D4
►On 16 points - transitive group 16T265
(1 5)(2 6)(3 7)(4 8)(9 15)(10 16)(11 13)(12 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16)(2 15)(3 14)(4 13)(5 10)(6 9)(7 12)(8 11)
(9 12 11 10)(13 16 15 14)
(1 7)(2 6)(3 5)(4 8)(9 14)(10 13)(11 16)(12 15)
G:=sub<Sym(16)| (1,5)(2,6)(3,7)(4,8)(9,15)(10,16)(11,13)(12,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,10)(6,9)(7,12)(8,11), (9,12,11,10)(13,16,15,14), (1,7)(2,6)(3,5)(4,8)(9,14)(10,13)(11,16)(12,15)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,15)(10,16)(11,13)(12,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,10)(6,9)(7,12)(8,11), (9,12,11,10)(13,16,15,14), (1,7)(2,6)(3,5)(4,8)(9,14)(10,13)(11,16)(12,15) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,15),(10,16),(11,13),(12,14)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16),(2,15),(3,14),(4,13),(5,10),(6,9),(7,12),(8,11)], [(9,12,11,10),(13,16,15,14)], [(1,7),(2,6),(3,5),(4,8),(9,14),(10,13),(11,16),(12,15)]])
G:=TransitiveGroup(16,265);
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4⋊4D4 |
| kernel | C2×D4⋊4D4 | C2×C4.D4 | C2×C4≀C2 | D4⋊4D4 | C2×C4⋊1D4 | C2×C8⋊C22 | C2×2+ 1+4 | C2×D4 | C2×Q8 | C4○D4 | C24 | C2 |
| # reps | 1 | 1 | 2 | 8 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 4 |
Matrix representation of C2×D4⋊4D4 ►in GL6(ℤ)
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0] >;
C2×D4⋊4D4 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes_4D_4
% in TeX
G:=Group("C2xD4:4D4"); // GroupNames label
G:=SmallGroup(128,1746);
// by ID
G=gap.SmallGroup(128,1746);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2804,1411,718,172,2028]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations