p-group, metabelian, nilpotent (class 3), monomial
Aliases: M5(2):12C22, C23.12M4(2), D4.C8:5C2, C4oD4.3C8, C8oD4.4C4, (C2xD4).8C8, D4.7(C2xC8), Q8.7(C2xC8), (C2xQ8).8C8, (C2xC8).209D4, C8.126(C2xD4), (C2xC16):10C22, C4.14(C22xC8), C8.32(C22:C4), C4.26(C22:C8), (C2xM5(2)):11C2, (C2xC8).604C23, C8oD4.16C22, (C2xC4).25M4(2), (C2xM4(2)).32C4, M4(2).34(C2xC4), C22.7(C22:C8), C22.2(C2xM4(2)), (C22xC8).417C22, (C2xC4).25(C2xC8), (C2xC8).150(C2xC4), C4oD4.31(C2xC4), (C2xC4oD4).20C4, (C2xC8oD4).19C2, C2.27(C2xC22:C8), C4.118(C2xC22:C4), (C2xC4).445(C22xC4), (C22xC4).287(C2xC4), (C2xC4).363(C22:C4), SmallGroup(128,849)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M5(2):12C22
G = < a,b,c,d | a16=b2=c2=d2=1, bab=dad=a9, cac=a5b, cbc=a8b, bd=db, cd=dc >
Subgroups: 172 in 110 conjugacy classes, 58 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C16, C2xC8, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C2xC16, C2xC16, M5(2), M5(2), C22xC8, C22xC8, C2xM4(2), C2xM4(2), C8oD4, C8oD4, C2xC4oD4, D4.C8, C2xM5(2), C2xC8oD4, M5(2):12C22
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, C23, C22:C4, C2xC8, M4(2), C22xC4, C2xD4, C22:C8, C2xC22:C4, C22xC8, C2xM4(2), C2xC22:C8, M5(2):12C22
►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 20)(3 29)(4 22)(5 31)(6 24)(7 17)(8 26)(9 19)(10 28)(11 21)(12 30)(13 23)(14 32)(15 25)(16 18)
(2 24)(4 26)(6 28)(8 30)(10 32)(12 18)(14 20)(16 22)(17 25)(19 27)(21 29)(23 31)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)
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)(2,20)(3,29)(4,22)(5,31)(6,24)(7,17)(8,26)(9,19)(10,28)(11,21)(12,30)(13,23)(14,32)(15,25)(16,18), (2,24)(4,26)(6,28)(8,30)(10,32)(12,18)(14,20)(16,22)(17,25)(19,27)(21,29)(23,31), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)>;
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)(2,20)(3,29)(4,22)(5,31)(6,24)(7,17)(8,26)(9,19)(10,28)(11,21)(12,30)(13,23)(14,32)(15,25)(16,18), (2,24)(4,26)(6,28)(8,30)(10,32)(12,18)(14,20)(16,22)(17,25)(19,27)(21,29)(23,31), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32) );
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),(2,20),(3,29),(4,22),(5,31),(6,24),(7,17),(8,26),(9,19),(10,28),(11,21),(12,30),(13,23),(14,32),(15,25),(16,18)], [(2,24),(4,26),(6,28),(8,30),(10,32),(12,18),(14,20),(16,22),(17,25),(19,27),(21,29),(23,31)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | D4 | M4(2) | M4(2) | M5(2):12C22 |
| kernel | M5(2):12C22 | D4.C8 | C2xM5(2) | C2xC8oD4 | C2xM4(2) | C8oD4 | C2xC4oD4 | C2xD4 | C2xQ8 | C4oD4 | C2xC8 | C2xC4 | C23 | C1 |
| # reps | 1 | 4 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 2 | 2 | 4 |
Matrix representation of M5(2):12C22 ►in GL4(F17) generated by
| 0 | 0 | 2 | 13 |
| 0 | 0 | 14 | 15 |
| 13 | 8 | 0 | 0 |
| 6 | 4 | 0 | 0 |
| 2 | 13 | 0 | 0 |
| 5 | 15 | 0 | 0 |
| 0 | 0 | 2 | 13 |
| 0 | 0 | 5 | 15 |
| 16 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 9 | 16 |
| 0 | 0 | 12 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,0,13,6,0,0,8,4,2,14,0,0,13,15,0,0],[2,5,0,0,13,15,0,0,0,0,2,5,0,0,13,15],[16,16,0,0,0,1,0,0,0,0,9,12,0,0,16,8],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16] >;
M5(2):12C22 in GAP, Magma, Sage, TeX
M_5(2)\rtimes_{12}C_2^2 % in TeX
G:=Group("M5(2):12C2^2"); // GroupNames label
G:=SmallGroup(128,849);
// by ID
G=gap.SmallGroup(128,849);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,1018,248,1411,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^2=d^2=1,b*a*b=d*a*d=a^9,c*a*c=a^5*b,c*b*c=a^8*b,b*d=d*b,c*d=d*c>;
// generators/relations