p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2):10D4, C42.376C23, (C2xC8):17D4, C8.3(C2xD4), C8:3D4:5C2, C8:5D4:5C2, (C4xC8):27C22, C4:C4.240D4, C8.2D4:5C2, C4:Q8:11C22, (C2xD8):51C22, C22:C4.80D4, C4.12(C22xD4), C8:C4:48C22, C8.12D4:15C2, C4.41(C4:1D4), (C2xC8).595C23, (C2xC4).352C24, (C2xQ16):21C22, (C22xSD16):5C2, C4.4D4:8C22, C23.454(C2xD4), C8o2M4(2):12C2, C2.35(D4oSD16), (C2xSD16):79C22, (C2xD4).118C23, C4:1D4.63C22, (C2xQ8).106C23, C22.29C24:12C2, C22.18(C4:1D4), (C22xC8).271C22, C22.612(C22xD4), (C22xC4).1042C23, (C22xD4).378C22, (C22xQ8).311C22, C42:C2.325C22, C23.38C23:12C2, (C2xM4(2)).272C22, (C2xC4oD8):21C2, (C2xC8:C22):24C2, (C2xC4).138(C2xD4), C2.31(C2xC4:1D4), (C2xC8.C22):24C2, (C2xC4oD4).158C22, SmallGroup(128,1886)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2):10D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a3, cbc-1=dbd=a4b, dcd=c-1 >
Subgroups: 604 in 282 conjugacy classes, 108 normal (34 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, C4:C4, C2xC8, C2xC8, M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C24, C4xC8, C8:C4, C42:C2, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C4:1D4, C4:Q8, C22xC8, C2xM4(2), C2xD8, C2xD8, C2xSD16, C2xSD16, C2xSD16, C2xQ16, C2xQ16, C4oD8, C8:C22, C8.C22, C22xD4, C22xQ8, C2xC4oD4, C8o2M4(2), C8:5D4, C8.12D4, C8:3D4, C8.2D4, C22.29C24, C23.38C23, C22xSD16, C2xC4oD8, C2xC8:C22, C2xC8.C22, M4(2):10D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C4:1D4, C22xD4, C2xC4:1D4, D4oSD16, M4(2):10D4
►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)(25 29)(27 31)
(1 9 32 23)(2 14 25 20)(3 11 26 17)(4 16 27 22)(5 13 28 19)(6 10 29 24)(7 15 30 21)(8 12 31 18)
(1 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 32)(10 27)(11 30)(12 25)(13 28)(14 31)(15 26)(16 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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31), (1,9,32,23)(2,14,25,20)(3,11,26,17)(4,16,27,22)(5,13,28,19)(6,10,29,24)(7,15,30,21)(8,12,31,18), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,32)(10,27)(11,30)(12,25)(13,28)(14,31)(15,26)(16,29)>;
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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31), (1,9,32,23)(2,14,25,20)(3,11,26,17)(4,16,27,22)(5,13,28,19)(6,10,29,24)(7,15,30,21)(8,12,31,18), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,32)(10,27)(11,30)(12,25)(13,28)(14,31)(15,26)(16,29) );
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)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31)], [(1,9,32,23),(2,14,25,20),(3,11,26,17),(4,16,27,22),(5,13,28,19),(6,10,29,24),(7,15,30,21),(8,12,31,18)], [(1,23),(2,18),(3,21),(4,24),(5,19),(6,22),(7,17),(8,20),(9,32),(10,27),(11,30),(12,25),(13,28),(14,31),(15,26),(16,29)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4oSD16 |
| kernel | M4(2):10D4 | C8o2M4(2) | C8:5D4 | C8.12D4 | C8:3D4 | C8.2D4 | C22.29C24 | C23.38C23 | C22xSD16 | C2xC4oD8 | C2xC8:C22 | C2xC8.C22 | C22:C4 | C4:C4 | C2xC8 | M4(2) | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of M4(2):10D4 ►in GL6(F17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 12 | 0 | 0 | 7 |
| 0 | 0 | 0 | 12 | 12 | 5 |
| 0 | 0 | 12 | 5 | 12 | 5 |
| 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 | 16 | 0 | 16 | 0 |
| 0 | 0 | 16 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 16 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 16 | 1 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,12,0,0,0,0,12,5,0,0,10,0,12,12,0,0,7,7,5,5],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,1,0,0,0,0,0,16,0,0,15,16,1,1,0,0,0,16,0,0] >;
M4(2):10D4 in GAP, Magma, Sage, TeX
M_4(2)\rtimes_{10}D_4 % in TeX
G:=Group("M4(2):10D4"); // GroupNames label
G:=SmallGroup(128,1886);
// by ID
G=gap.SmallGroup(128,1886);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,184,521,2804,172]);
// 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^3,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations