p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊19D4, C4⋊Q8⋊14C4, C4.85(C4×D4), C4⋊1D4⋊10C4, (C2×D4).72D4, C4.4D4⋊11C4, Q8○M4(2)⋊6C2, (C22×C4).63D4, C42⋊6C4⋊21C2, C4.126C22≀C2, C42.147(C2×C4), C23.127(C2×D4), C22.2(C4⋊D4), C23.8(C22⋊C4), C42⋊C22⋊12C2, C23.C23⋊4C2, (C22×C4).682C23, (C2×C42).283C22, C42⋊C2.19C22, C4.108(C22.D4), C2.23(C23.23D4), (C2×M4(2)).179C22, C22.26C24.19C2, (C2×D4).75(C2×C4), (C2×Q8).66(C2×C4), (C2×C4).54(C4○D4), (C2×C4).1003(C2×D4), (C2×C4).12(C22⋊C4), (C2×C4).184(C22×C4), (C2×C4○D4).19C22, C22.39(C2×C22⋊C4), SmallGroup(128,616)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)⋊19D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a3b, dad=a-1b, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 340 in 166 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C23⋊C4, C4≀C2, C2×C42, C42⋊C2, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C42⋊6C4, C23.C23, C42⋊C22, C22.26C24, Q8○M4(2), M4(2)⋊19D4
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, M4(2)⋊19D4
►On 16 points - transitive group 16T251
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(10 14)(12 16)
(2 4 6 8)(9 15 13 11)
(1 10)(2 9)(3 12)(4 11)(5 14)(6 13)(7 16)(8 15)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (2,4,6,8)(9,15,13,11), (1,10)(2,9)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (2,4,6,8)(9,15,13,11), (1,10)(2,9)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(10,14),(12,16)], [(2,4,6,8),(9,15,13,11)], [(1,10),(2,9),(3,12),(4,11),(5,14),(6,13),(7,16),(8,15)]])
G:=TransitiveGroup(16,251);
►On 16 points - transitive group 16T299
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)
(1 14 5 10)(3 16 7 12)
(1 5)(2 15)(3 7)(4 9)(6 11)(8 13)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11), (1,14,5,10)(3,16,7,12), (1,5)(2,15)(3,7)(4,9)(6,11)(8,13)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11), (1,14,5,10)(3,16,7,12), (1,5)(2,15)(3,7)(4,9)(6,11)(8,13) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,16),(2,13),(3,10),(4,15),(5,12),(6,9),(7,14),(8,11)], [(1,14,5,10),(3,16,7,12)], [(1,5),(2,15),(3,7),(4,9),(6,11),(8,13)]])
G:=TransitiveGroup(16,299);
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4K | 4L | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D4 | C4○D4 | M4(2)⋊19D4 |
| kernel | M4(2)⋊19D4 | C42⋊6C4 | C23.C23 | C42⋊C22 | C22.26C24 | Q8○M4(2) | C4.4D4 | C4⋊1D4 | C4⋊Q8 | M4(2) | C22×C4 | C2×D4 | C2×C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of M4(2)⋊19D4 ►in GL4(𝔽5) generated by
| 0 | 0 | 0 | 2 |
| 0 | 0 | 1 | 0 |
| 0 | 3 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,0,0,4,0,0,3,0,0,1,0,0,2,0,0,0],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,2,0,0,0,0,1,0,0,0,0,3],[0,0,3,0,0,0,0,4,2,0,0,0,0,4,0,0] >;
M4(2)⋊19D4 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_{19}D_4 % in TeX
G:=Group("M4(2):19D4"); // GroupNames label
G:=SmallGroup(128,616);
// by ID
G=gap.SmallGroup(128,616);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,521,248,2804,1411,2028,1027]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,c*a*c^-1=a^3*b,d*a*d=a^-1*b,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations