p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).31D4, C4.91(C4×D4), C4⋊C4.222D4, (C2×SD16)⋊8C4, (C2×C8).114D4, C22.62(C4×D4), C4.20(C4⋊1D4), C4.87(C4⋊D4), C8.27(C22⋊C4), C8○2M4(2)⋊2C2, C2.7(D4.3D4), (C22×SD16).2C2, C23.269(C4○D4), C23.38D4⋊26C2, (C22×C8).226C22, (C22×C4).703C23, C23.37D4.6C2, (C22×D4).52C22, (C22×Q8).42C22, C22.149(C4⋊D4), C22.18(C4.4D4), C42⋊C2.278C22, (C2×M4(2)).213C22, C2.29(C24.3C22), (C2×C8).76(C2×C4), (C2×C4).33(C2×D4), C4.45(C2×C22⋊C4), (C2×C8.C4)⋊12C2, (C2×D4).116(C2×C4), (C2×C4).67(C4○D4), (C2×Q8).101(C2×C4), (C2×C4.10D4)⋊23C2, (C2×C4).201(C22×C4), (C2×C4.D4).10C2, SmallGroup(128,709)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).31D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6b, bab=a5, ac=ca, dad-1=a5b, bc=cb, dbd-1=a4b, dcd-1=c3 >
Subgroups: 332 in 148 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4×C8, C8⋊C4, C4.D4, C4.10D4, D4⋊C4, Q8⋊C4, C8.C4, C42⋊C2, C22×C8, C2×M4(2), C2×SD16, C2×SD16, C22×D4, C22×Q8, C8○2M4(2), C2×C4.D4, C2×C4.10D4, C23.37D4, C23.38D4, C2×C8.C4, C22×SD16, M4(2).31D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C24.3C22, D4.3D4, M4(2).31D4
►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 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)
(1 17 29 9 5 21 25 13)(2 18 30 10 6 22 26 14)(3 19 31 11 7 23 27 15)(4 20 32 12 8 24 28 16)
(1 18 25 10 5 22 29 14)(2 9 30 17 6 13 26 21)(3 24 27 16 7 20 31 12)(4 15 32 23 8 11 28 19)
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), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22), (1,17,29,9,5,21,25,13)(2,18,30,10,6,22,26,14)(3,19,31,11,7,23,27,15)(4,20,32,12,8,24,28,16), (1,18,25,10,5,22,29,14)(2,9,30,17,6,13,26,21)(3,24,27,16,7,20,31,12)(4,15,32,23,8,11,28,19)>;
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), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22), (1,17,29,9,5,21,25,13)(2,18,30,10,6,22,26,14)(3,19,31,11,7,23,27,15)(4,20,32,12,8,24,28,16), (1,18,25,10,5,22,29,14)(2,9,30,17,6,13,26,21)(3,24,27,16,7,20,31,12)(4,15,32,23,8,11,28,19) );
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)], [(1,27),(2,32),(3,29),(4,26),(5,31),(6,28),(7,25),(8,30),(9,19),(10,24),(11,21),(12,18),(13,23),(14,20),(15,17),(16,22)], [(1,17,29,9,5,21,25,13),(2,18,30,10,6,22,26,14),(3,19,31,11,7,23,27,15),(4,20,32,12,8,24,28,16)], [(1,18,25,10,5,22,29,14),(2,9,30,17,6,13,26,21),(3,24,27,16,7,20,31,12),(4,15,32,23,8,11,28,19)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | C4○D4 | C4○D4 | D4.3D4 |
| kernel | M4(2).31D4 | C8○2M4(2) | C2×C4.D4 | C2×C4.10D4 | C23.37D4 | C23.38D4 | C2×C8.C4 | C22×SD16 | C2×SD16 | C4⋊C4 | C2×C8 | M4(2) | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of M4(2).31D4 ►in GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 11 | 4 | 0 | 0 | 0 | 0 |
| 12 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 6 | 13 | 0 | 0 | 0 | 0 |
| 13 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,12,0,0,0,0,4,6,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,5,12],[6,13,0,0,0,0,13,11,0,0,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,5,5,0,0,0,0,5,12,0,0] >;
M4(2).31D4 in GAP, Magma, Sage, TeX
M_4(2)._{31}D_4 % in TeX
G:=Group("M4(2).31D4"); // GroupNames label
G:=SmallGroup(128,709);
// by ID
G=gap.SmallGroup(128,709);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,1018,2804,172,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^6*b,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^5*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;
// generators/relations