p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.3C42, C23.6Q16, M5(2).2C4, (C2xC8).7Q8, C8.14(C4:C4), (C2xC4).124D8, (C2xC8).354D4, C4.4(C4.Q8), C8.C4.5C4, (C2xC4).23SD16, C8.43(C22:C4), (C22xC4).194D4, C4.51(D4:C4), C22.4(C2.D8), (C2xM5(2)).16C2, C4.9(C2.C42), (C22xC8).207C22, C22.4(Q8:C4), C2.18(C22.4Q16), (C2xC8).54(C2xC4), (C2xC4).30(C4:C4), (C2xC8.C4).6C2, (C2xC4).232(C22:C4), SmallGroup(128,120)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M5(2).C4
G = < a,b,c | a16=b2=1, c4=a8, bab=a9, cac-1=a7b, cbc-1=a8b >
Subgroups: 104 in 62 conjugacy classes, 38 normal (32 characteristic)
C1, C2, C2, C4, C22, C22, C8, C8, C2xC4, C23, C16, C2xC8, C2xC8, M4(2), C22xC4, C8.C4, C8.C4, C2xC16, M5(2), M5(2), C22xC8, C2xM4(2), C2xC8.C4, C2xM5(2), M5(2).C4
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C42, C22:C4, C4:C4, D8, SD16, Q16, C2.C42, D4:C4, Q8:C4, C4.Q8, C2.D8, C22.4Q16, M5(2).C4
►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 9)(3 11)(5 13)(7 15)(17 25)(19 27)(21 29)(23 31)
(1 28 13 32 9 20 5 24)(2 27 14 31 10 19 6 23)(3 18 15 22 11 26 7 30)(4 17 16 21 12 25 8 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), (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31), (1,28,13,32,9,20,5,24)(2,27,14,31,10,19,6,23)(3,18,15,22,11,26,7,30)(4,17,16,21,12,25,8,29)>;
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), (1,9)(3,11)(5,13)(7,15)(17,25)(19,27)(21,29)(23,31), (1,28,13,32,9,20,5,24)(2,27,14,31,10,19,6,23)(3,18,15,22,11,26,7,30)(4,17,16,21,12,25,8,29) );
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)], [(1,9),(3,11),(5,13),(7,15),(17,25),(19,27),(21,29),(23,31)], [(1,28,13,32,9,20,5,24),(2,27,14,31,10,19,6,23),(3,18,15,22,11,26,7,30),(4,17,16,21,12,25,8,29)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | ··· | 8N | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D4 | D8 | SD16 | Q16 | M5(2).C4 |
| kernel | M5(2).C4 | C2xC8.C4 | C2xM5(2) | C8.C4 | M5(2) | C2xC8 | C2xC8 | C22xC4 | C2xC4 | C2xC4 | C23 | C1 |
| # reps | 1 | 2 | 1 | 8 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 4 |
Matrix representation of M5(2).C4 ►in GL4(F17) generated by
| 0 | 16 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
G:=sub<GL(4,GF(17))| [0,15,0,0,16,0,0,0,0,0,0,1,0,0,8,0],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[0,0,13,0,0,0,0,13,1,0,0,0,0,1,0,0] >;
M5(2).C4 in GAP, Magma, Sage, TeX
M_5(2).C_4
% in TeX
G:=Group("M5(2).C4"); // GroupNames label
G:=SmallGroup(128,120);
// by ID
G=gap.SmallGroup(128,120);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,723,520,248,1684,102,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^2=1,c^4=a^8,b*a*b=a^9,c*a*c^-1=a^7*b,c*b*c^-1=a^8*b>;
// generators/relations