p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).40D4, C4.52(C4×D4), (C2×D4).6Q8, C23.3(C2×Q8), C23.5(C4⋊C4), C4.D4.2C4, (C22×C4).61D4, C4.117C22≀C2, C4.C42⋊2C2, C4.10D4.2C4, Q8○M4(2).1C2, M4(2).2(C2×C4), M4(2).C4⋊1C2, (C22×C8).27C22, (C22×C4).674C23, C22.15(C22⋊Q8), C2.14(C23.8Q8), C4.106(C22.D4), (C2×M4(2)).172C22, M4(2).8C22.1C2, (C2×C4).6(C4⋊C4), (C2×D4).67(C2×C4), (C2×C4).984(C2×D4), C22.24(C2×C4⋊C4), (C2×C4).8(C22×C4), (C2×Q8).60(C2×C4), (C2×C4).317(C4○D4), (C2×C4○D4).10C22, (C22×C8)⋊C2.13C2, SmallGroup(128,590)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).40D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6, bab=a5, cac-1=a3b, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=a2c3 >
Subgroups: 212 in 126 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C22⋊C8, C4.D4, C4.10D4, C8.C4, C22×C8, C2×M4(2), C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C4.C42, (C22×C8)⋊C2, M4(2).8C22, M4(2).C4, Q8○M4(2), M4(2).40D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C23.8Q8, M4(2).40D4
►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 17)(2 22)(3 19)(4 24)(5 21)(6 18)(7 23)(8 20)(9 30)(10 27)(11 32)(12 29)(13 26)(14 31)(15 28)(16 25)
(1 28 7 26 5 32 3 30)(2 10 4 12 6 14 8 16)(9 17 15 23 13 21 11 19)(18 31 20 25 22 27 24 29)
(1 28 7 26 5 32 3 30)(2 10 8 16 6 14 4 12)(9 21 15 19 13 17 11 23)(18 27 24 25 22 31 20 29)
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), (1,17)(2,22)(3,19)(4,24)(5,21)(6,18)(7,23)(8,20)(9,30)(10,27)(11,32)(12,29)(13,26)(14,31)(15,28)(16,25), (1,28,7,26,5,32,3,30)(2,10,4,12,6,14,8,16)(9,17,15,23,13,21,11,19)(18,31,20,25,22,27,24,29), (1,28,7,26,5,32,3,30)(2,10,8,16,6,14,4,12)(9,21,15,19,13,17,11,23)(18,27,24,25,22,31,20,29)>;
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), (1,17)(2,22)(3,19)(4,24)(5,21)(6,18)(7,23)(8,20)(9,30)(10,27)(11,32)(12,29)(13,26)(14,31)(15,28)(16,25), (1,28,7,26,5,32,3,30)(2,10,4,12,6,14,8,16)(9,17,15,23,13,21,11,19)(18,31,20,25,22,27,24,29), (1,28,7,26,5,32,3,30)(2,10,8,16,6,14,4,12)(9,21,15,19,13,17,11,23)(18,27,24,25,22,31,20,29) );
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)], [(1,17),(2,22),(3,19),(4,24),(5,21),(6,18),(7,23),(8,20),(9,30),(10,27),(11,32),(12,29),(13,26),(14,31),(15,28),(16,25)], [(1,28,7,26,5,32,3,30),(2,10,4,12,6,14,8,16),(9,17,15,23,13,21,11,19),(18,31,20,25,22,27,24,29)], [(1,28,7,26,5,32,3,30),(2,10,8,16,6,14,4,12),(9,21,15,19,13,17,11,23),(18,27,24,25,22,31,20,29)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | ··· | 8L | 8M | ··· | 8R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | Q8 | C4○D4 | M4(2).40D4 |
| kernel | M4(2).40D4 | C4.C42 | (C22×C8)⋊C2 | M4(2).8C22 | M4(2).C4 | Q8○M4(2) | C4.D4 | C4.10D4 | M4(2) | C22×C4 | C2×D4 | C2×C4 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of M4(2).40D4 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 8 |
| 16 | 0 | 4 | 4 |
| 10 | 1 | 0 | 1 |
| 11 | 0 | 0 | 1 |
| 13 | 8 | 0 | 0 |
| 13 | 4 | 0 | 0 |
| 16 | 1 | 0 | 4 |
| 0 | 1 | 13 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 15 | 0 | 8 | 0 |
| 9 | 0 | 0 | 0 |
| 9 | 8 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(17))| [16,16,10,11,0,0,1,0,0,4,0,0,8,4,1,1],[13,13,16,0,8,4,1,1,0,0,0,13,0,0,4,0],[9,0,0,15,0,9,0,0,0,0,0,8,0,0,9,0],[9,9,0,0,0,8,0,0,0,0,0,9,0,0,9,0] >;
M4(2).40D4 in GAP, Magma, Sage, TeX
M_4(2)._{40}D_4 % in TeX
G:=Group("M4(2).40D4"); // GroupNames label
G:=SmallGroup(128,590);
// by ID
G=gap.SmallGroup(128,590);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,2019,1018,521,248,1411,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^3*b,d*a*d^-1=a^-1*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^3>;
// generators/relations