p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊20D4, C24.8(C2×C4), C4.93C22≀C2, C22.51(C4×D4), (C22×D4).13C4, (C22×C4).294D4, C4.133(C4⋊D4), C24.4C4⋊29C2, C22⋊1(C4.D4), C22.C42⋊17C2, (C22×M4(2))⋊12C2, (C22×C4).688C23, C23.194(C22×C4), (C23×C4).263C22, (C22×D4).28C22, C23.125(C22⋊C4), C4.88(C22.D4), C2.39(C23.23D4), (C2×M4(2)).187C22, C2.26(M4(2).8C22), (C2×C4⋊D4).10C2, (C2×C4.D4)⋊17C2, (C2×C4).1336(C2×D4), (C2×C4⋊C4).64C22, (C2×C22⋊C4).11C4, (C22×C4).20(C2×C4), C2.27(C2×C4.D4), (C2×C4).322(C4○D4), (C2×C4).197(C22⋊C4), C22.275(C2×C22⋊C4), SmallGroup(128,632)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2)⋊20D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=dad=ab, bc=cb, bd=db, dcd=c-1 >
Subgroups: 444 in 195 conjugacy classes, 58 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C22⋊C8, C4.D4, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C22×D4, C22×D4, C22.C42, C24.4C4, C2×C4.D4, C2×C4⋊D4, C22×M4(2), M4(2)⋊20D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C4.D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C2×C4.D4, M4(2).8C22, M4(2)⋊20D4
►On 32 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)
(1 5)(3 7)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)
(1 12 31 19)(2 13 32 20)(3 10 25 17)(4 11 26 18)(5 16 27 23)(6 9 28 24)(7 14 29 21)(8 15 30 22)
(3 7)(4 8)(9 24)(10 21)(11 22)(12 19)(13 20)(14 17)(15 18)(16 23)(25 29)(26 30)
G:=sub<Sym(32)| (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), (1,5)(3,7)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31), (1,12,31,19)(2,13,32,20)(3,10,25,17)(4,11,26,18)(5,16,27,23)(6,9,28,24)(7,14,29,21)(8,15,30,22), (3,7)(4,8)(9,24)(10,21)(11,22)(12,19)(13,20)(14,17)(15,18)(16,23)(25,29)(26,30)>;
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), (1,5)(3,7)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31), (1,12,31,19)(2,13,32,20)(3,10,25,17)(4,11,26,18)(5,16,27,23)(6,9,28,24)(7,14,29,21)(8,15,30,22), (3,7)(4,8)(9,24)(10,21)(11,22)(12,19)(13,20)(14,17)(15,18)(16,23)(25,29)(26,30) );
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)], [(1,5),(3,7),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31)], [(1,12,31,19),(2,13,32,20),(3,10,25,17),(4,11,26,18),(5,16,27,23),(6,9,28,24),(7,14,29,21),(8,15,30,22)], [(3,7),(4,8),(9,24),(10,21),(11,22),(12,19),(13,20),(14,17),(15,18),(16,23),(25,29),(26,30)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | ··· | 4F | 4G | 4H | 4I | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D4 | C4.D4 | M4(2).8C22 |
| kernel | M4(2)⋊20D4 | C22.C42 | C24.4C4 | C2×C4.D4 | C2×C4⋊D4 | C22×M4(2) | C2×C22⋊C4 | C22×D4 | M4(2) | C22×C4 | C2×C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of M4(2)⋊20D4 ►in GL6(𝔽17)
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 8 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 9 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,16,0,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,8,4,0,0,0,0,0,0,4,0,0,0,0,0,9,13],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,1] >;
M4(2)⋊20D4 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_{20}D_4 % in TeX
G:=Group("M4(2):20D4"); // GroupNames label
G:=SmallGroup(128,632);
// by ID
G=gap.SmallGroup(128,632);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,1018,2028,124]);
// 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=d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations