p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2):15D4, C24.112D4, C8:D4:2C2, C8.22(C2xD4), C8:8D4:12C2, C8.D4:2C2, C4.Q8:4C22, C4:C4.25C23, C2.D8:15C22, C8.18D4:25C2, (C2xC4).260C24, (C2xC8).252C23, (C2xQ16):18C22, (C2xD4).63C23, C4.154(C22xD4), C23.385(C2xD4), (C22xC4).430D4, (C2xQ8).51C23, C4.173(C4:D4), D4:C4:51C22, Q8:C4:52C22, (C2xSD16):12C22, C22:1(C8.C22), (C22xM4(2)):5C2, M4(2):C4:13C2, C23.36D4:41C2, C23.38D4:33C2, C4:D4.150C22, C22.85(C4:D4), (C23xC4).552C22, (C22xC8).259C22, (C22xC4).982C23, C22.520(C22xD4), C22:Q8.155C22, C2.16(D8:C22), C22.19C24.16C2, (C22xQ8).281C22, C42:C2.109C22, (C2xM4(2)).264C22, C4.27(C2xC4oD4), (C2xC4).476(C2xD4), C2.78(C2xC4:D4), (C2xC22:Q8):56C2, (C2xC8.C22):18C2, C2.19(C2xC8.C22), (C2xC4).478(C4oD4), (C2xC4:C4).592C22, (C2xC4oD4).126C22, SmallGroup(128,1788)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2):15D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=dad=a3, cbc-1=dbd=a4b, dcd=c-1 >
Subgroups: 460 in 248 conjugacy classes, 102 normal (44 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), M4(2), SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C24, D4:C4, Q8:C4, C4.Q8, C2.D8, C2xC22:C4, C2xC4:C4, C2xC4:C4, C42:C2, C4xD4, C22wrC2, C4:D4, C22:Q8, C22:Q8, C22:Q8, C22.D4, C22xC8, C2xM4(2), C2xM4(2), C2xM4(2), C2xSD16, C2xQ16, C8.C22, C23xC4, C22xQ8, C2xC4oD4, C23.36D4, C23.38D4, M4(2):C4, C8:8D4, C8.18D4, C8:D4, C8.D4, C2xC22:Q8, C22.19C24, C22xM4(2), C2xC8.C22, M4(2):15D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C4:D4, C8.C22, C22xD4, C2xC4oD4, C2xC4:D4, C2xC8.C22, D8:C22, M4(2):15D4
►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 23)(2 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)
(1 13 23 31)(2 16 24 26)(3 11 17 29)(4 14 18 32)(5 9 19 27)(6 12 20 30)(7 15 21 25)(8 10 22 28)
(1 31)(2 26)(3 29)(4 32)(5 27)(6 30)(7 25)(8 28)(9 19)(10 22)(11 17)(12 20)(13 23)(14 18)(15 21)(16 24)
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), (1,23)(2,20)(3,17)(4,22)(5,19)(6,24)(7,21)(8,18)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26), (1,13,23,31)(2,16,24,26)(3,11,17,29)(4,14,18,32)(5,9,19,27)(6,12,20,30)(7,15,21,25)(8,10,22,28), (1,31)(2,26)(3,29)(4,32)(5,27)(6,30)(7,25)(8,28)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24)>;
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), (1,23)(2,20)(3,17)(4,22)(5,19)(6,24)(7,21)(8,18)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26), (1,13,23,31)(2,16,24,26)(3,11,17,29)(4,14,18,32)(5,9,19,27)(6,12,20,30)(7,15,21,25)(8,10,22,28), (1,31)(2,26)(3,29)(4,32)(5,27)(6,30)(7,25)(8,28)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24) );
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)], [(1,23),(2,20),(3,17),(4,22),(5,19),(6,24),(7,21),(8,18),(9,31),(10,28),(11,25),(12,30),(13,27),(14,32),(15,29),(16,26)], [(1,13,23,31),(2,16,24,26),(3,11,17,29),(4,14,18,32),(5,9,19,27),(6,12,20,30),(7,15,21,25),(8,10,22,28)], [(1,31),(2,26),(3,29),(4,32),(5,27),(6,30),(7,25),(8,28),(9,19),(10,22),(11,17),(12,20),(13,23),(14,18),(15,21),(16,24)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | ··· | 2 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4oD4 | C8.C22 | D8:C22 |
| kernel | M4(2):15D4 | C23.36D4 | C23.38D4 | M4(2):C4 | C8:8D4 | C8.18D4 | C8:D4 | C8.D4 | C2xC22:Q8 | C22.19C24 | C22xM4(2) | C2xC8.C22 | M4(2) | C22xC4 | C24 | C2xC4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 1 | 4 | 2 | 2 |
Matrix representation of M4(2):15D4 ►in GL6(F17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 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 | 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,10,0,0,0,0,12,0,0,0,0,7,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,10,0,0,0,0,12,0,0,0,0,7,0,0,0] >;
M4(2):15D4 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_{15}D_4 % in TeX
G:=Group("M4(2):15D4"); // GroupNames label
G:=SmallGroup(128,1788);
// by ID
G=gap.SmallGroup(128,1788);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,2804,172]);
// 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^3,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations