p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2):11D4, C42.377C23, (C2xC8):18D4, C8.4(C2xD4), C8:3D4:6C2, C8:4D4:18C2, C4:C4.241D4, (C4xC8):28C22, C2.22(D4oD8), (C22xD8):19C2, (C2xD8):21C22, C4:1D4:7C22, C22:C4.81D4, C4.13(C22xD4), C8:C4:49C22, C8.12D4:16C2, C4.42(C4:1D4), (C2xC4).353C24, (C2xC8).560C23, (C2xQ16):45C22, C4.4D4:9C22, C23.455(C2xD4), C8o2M4(2):13C2, (C2xSD16):17C22, (C2xD4).119C23, (C2xQ8).107C23, C22.29C24:13C2, C22.19(C4:1D4), (C22xC8).272C22, C22.613(C22xD4), (C22xC4).1043C23, (C22xD4).379C22, C42:C2.326C22, (C2xM4(2)).273C22, (C2xC4oD8):22C2, (C2xC8:C22):25C2, (C2xC4).139(C2xD4), C2.32(C2xC4:1D4), (C2xC4oD4).159C22, SmallGroup(128,1887)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2):11D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a-1, cbc-1=dbd=a4b, dcd=c-1 >
Subgroups: 716 in 298 conjugacy classes, 108 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C22:C4, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C4xC8, C8:C4, C42:C2, C22wrC2, C4:D4, C4.4D4, C4:1D4, C22xC8, C2xM4(2), C2xD8, C2xD8, C2xD8, C2xSD16, C2xQ16, C4oD8, C8:C22, C22xD4, C2xC4oD4, C8o2M4(2), C8:4D4, C8.12D4, C8:3D4, C22.29C24, C22xD8, C2xC4oD8, C2xC8:C22, M4(2):11D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C4:1D4, C22xD4, C2xC4:1D4, D4oD8, M4(2):11D4
►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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)
(1 11 29 19)(2 16 30 24)(3 13 31 21)(4 10 32 18)(5 15 25 23)(6 12 26 20)(7 9 27 17)(8 14 28 22)
(1 28)(2 27)(3 26)(4 25)(5 32)(6 31)(7 30)(8 29)(9 16)(10 15)(11 14)(12 13)(17 24)(18 23)(19 22)(20 21)
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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,11,29,19)(2,16,30,24)(3,13,31,21)(4,10,32,18)(5,15,25,23)(6,12,26,20)(7,9,27,17)(8,14,28,22), (1,28)(2,27)(3,26)(4,25)(5,32)(6,31)(7,30)(8,29)(9,16)(10,15)(11,14)(12,13)(17,24)(18,23)(19,22)(20,21)>;
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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,11,29,19)(2,16,30,24)(3,13,31,21)(4,10,32,18)(5,15,25,23)(6,12,26,20)(7,9,27,17)(8,14,28,22), (1,28)(2,27)(3,26)(4,25)(5,32)(6,31)(7,30)(8,29)(9,16)(10,15)(11,14)(12,13)(17,24)(18,23)(19,22)(20,21) );
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)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32)], [(1,11,29,19),(2,16,30,24),(3,13,31,21),(4,10,32,18),(5,15,25,23),(6,12,26,20),(7,9,27,17),(8,14,28,22)], [(1,28),(2,27),(3,26),(4,25),(5,32),(6,31),(7,30),(8,29),(9,16),(10,15),(11,14),(12,13),(17,24),(18,23),(19,22),(20,21)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| 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 |
| 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 |
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 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4oD8 |
| kernel | M4(2):11D4 | C8o2M4(2) | C8:4D4 | C8.12D4 | C8:3D4 | C22.29C24 | C22xD8 | C2xC4oD8 | C2xC8:C22 | C22:C4 | C4:C4 | C2xC8 | M4(2) | C2 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of M4(2):11D4 ►in GL6(F17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 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 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,14,14,0,0,0,0,3,14,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,14,14,0,0,0,0,14,3,0,0,14,14,0,0,0,0,14,3,0,0] >;
M4(2):11D4 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_{11}D_4 % in TeX
G:=Group("M4(2):11D4"); // GroupNames label
G:=SmallGroup(128,1887);
// by ID
G=gap.SmallGroup(128,1887);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,184,521,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^5,d*a*d=a^-1,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations