p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).32D4, (C2×D8)⋊13C4, C4.92(C4×D4), C4⋊C4.223D4, (C2×C8).115D4, C22.63(C4×D4), C4.88(C4⋊D4), C4.21(C4⋊1D4), C8.22(C22⋊C4), C8○2M4(2)⋊3C2, (C22×D8).12C2, C2.5(D4.4D4), C23.270(C4○D4), C23.37D4⋊26C2, (C22×C8).227C22, (C22×C4).704C23, (C22×D4).53C22, C22.150(C4⋊D4), C22.19(C4.4D4), C42⋊C2.279C22, (C2×M4(2)).214C22, C2.30(C24.3C22), (C2×C8).77(C2×C4), (C2×C4).34(C2×D4), C4.46(C2×C22⋊C4), (C2×C8.C4)⋊13C2, (C2×D4).117(C2×C4), (C2×C4).68(C4○D4), (C2×C4.D4)⋊23C2, (C2×C4).202(C22×C4), SmallGroup(128,710)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).32D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2b, bab=a5, ac=ca, dad-1=a5b, bc=cb, dbd-1=a4b, dcd-1=a4c3 >
Subgroups: 412 in 158 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), D8, C22×C4, C2×D4, C2×D4, C24, C4×C8, C8⋊C4, C4.D4, D4⋊C4, C8.C4, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C2×D8, C2×D8, C22×D4, C8○2M4(2), C2×C4.D4, C23.37D4, C2×C8.C4, C22×D8, M4(2).32D4
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, D4.4D4, M4(2).32D4
►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 21 25 9 5 17 29 13)(2 22 26 10 6 18 30 14)(3 23 27 11 7 19 31 15)(4 24 28 12 8 20 32 16)
(1 2 29 26 5 6 25 30)(3 8 31 32 7 4 27 28)(9 18 21 14 13 22 17 10)(11 24 23 12 15 20 19 16)
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), (1,21,25,9,5,17,29,13)(2,22,26,10,6,18,30,14)(3,23,27,11,7,19,31,15)(4,24,28,12,8,20,32,16), (1,2,29,26,5,6,25,30)(3,8,31,32,7,4,27,28)(9,18,21,14,13,22,17,10)(11,24,23,12,15,20,19,16)>;
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), (1,21,25,9,5,17,29,13)(2,22,26,10,6,18,30,14)(3,23,27,11,7,19,31,15)(4,24,28,12,8,20,32,16), (1,2,29,26,5,6,25,30)(3,8,31,32,7,4,27,28)(9,18,21,14,13,22,17,10)(11,24,23,12,15,20,19,16) );
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)], [(1,21,25,9,5,17,29,13),(2,22,26,10,6,18,30,14),(3,23,27,11,7,19,31,15),(4,24,28,12,8,20,32,16)], [(1,2,29,26,5,6,25,30),(3,8,31,32,7,4,27,28),(9,18,21,14,13,22,17,10),(11,24,23,12,15,20,19,16)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | C4○D4 | C4○D4 | D4.4D4 |
| kernel | M4(2).32D4 | C8○2M4(2) | C2×C4.D4 | C23.37D4 | C2×C8.C4 | C22×D8 | C2×D8 | C4⋊C4 | C2×C8 | M4(2) | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 2 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of M4(2).32D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 6 | 3 | 1 | 15 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 5 | 13 | 14 |
| 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 | 16 | 0 |
| 0 | 0 | 6 | 3 | 0 | 16 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 11 |
| 0 | 0 | 13 | 5 | 3 | 11 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 6 | 3 | 1 | 15 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 4 | 7 | 14 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,6,0,1,0,0,0,3,1,5,0,0,1,1,0,13,0,0,0,15,0,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,6,0,0,0,1,0,3,0,0,0,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,14,14,1,13,0,0,3,14,9,5,0,0,0,0,0,3,0,0,0,0,11,11],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,1,0,0,0,3,16,4,0,0,16,1,0,7,0,0,0,15,0,14] >;
M4(2).32D4 in GAP, Magma, Sage, TeX
M_4(2)._{32}D_4 % in TeX
G:=Group("M4(2).32D4"); // GroupNames label
G:=SmallGroup(128,710);
// by ID
G=gap.SmallGroup(128,710);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,436,2019,1018,2804,172,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^2*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=a^4*c^3>;
// generators/relations