p-group, metabelian, nilpotent (class 4), monomial
Aliases: M5(2).1C4, C23.10Q16, C16.7(C2×C4), C4.91(C2×D8), C8.11(C4⋊C4), (C2×C8).21Q8, C8.25(C2×Q8), (C2×C4).146D8, (C2×C8).129D4, C8.4Q8⋊3C2, (C2×C4).25Q16, C4.7(C2.D8), C8.56(C22×C4), C22.2(C2×Q16), (C2×C8).579C23, (C2×C16).18C22, (C22×C4).342D4, (C2×M5(2)).2C2, C22.7(C2.D8), C8.C4.15C22, (C22×C8).241C22, C4.55(C2×C4⋊C4), (C2×C8).92(C2×C4), C2.16(C2×C2.D8), (C2×C4).59(C4⋊C4), (C2×C4).767(C2×D4), (C2×C8.C4).25C2, SmallGroup(128,893)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M5(2).1C4
G = < a,b,c | a16=b2=1, c4=a8, bab=a9, cac-1=a7, cbc-1=a8b >
Subgroups: 108 in 70 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C8, C2×C4, C2×C4, C23, C16, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C8.C4, C8.C4, C2×C16, M5(2), C22×C8, C2×M4(2), C8.4Q8, C2×C8.C4, C2×M5(2), M5(2).1C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C2×C2.D8, M5(2).1C4
►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 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)
(1 31 13 19 9 23 5 27)(2 22 14 26 10 30 6 18)(3 29 15 17 11 21 7 25)(4 20 16 24 12 28 8 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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31), (1,31,13,19,9,23,5,27)(2,22,14,26,10,30,6,18)(3,29,15,17,11,21,7,25)(4,20,16,24,12,28,8,32)>;
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), (2,10)(4,12)(6,14)(8,16)(17,25)(19,27)(21,29)(23,31), (1,31,13,19,9,23,5,27)(2,22,14,26,10,30,6,18)(3,29,15,17,11,21,7,25)(4,20,16,24,12,28,8,32) );
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)], [(2,10),(4,12),(6,14),(8,16),(17,25),(19,27),(21,29),(23,31)], [(1,31,13,19,9,23,5,27),(2,22,14,26,10,30,6,18),(3,29,15,17,11,21,7,25),(4,20,16,24,12,28,8,32)]])
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 | C2 | C4 | D4 | Q8 | D4 | D8 | Q16 | Q16 | M5(2).1C4 |
| kernel | M5(2).1C4 | C8.4Q8 | C2×C8.C4 | C2×M5(2) | M5(2) | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C1 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of M5(2).1C4 ►in GL4(𝔽17) generated by
| 2 | 16 | 16 | 14 |
| 13 | 15 | 15 | 15 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 4 | 0 |
| 1 | 1 | 1 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 10 | 10 |
| 0 | 0 | 0 | 1 |
| 9 | 13 | 13 | 13 |
| 0 | 4 | 0 | 0 |
G:=sub<GL(4,GF(17))| [2,13,0,0,16,15,0,0,16,15,0,4,14,15,8,0],[1,0,0,0,1,16,0,0,1,0,16,0,0,0,0,1],[4,0,9,0,0,0,13,4,10,0,13,0,10,1,13,0] >;
M5(2).1C4 in GAP, Magma, Sage, TeX
M_5(2)._1C_4
% in TeX
G:=Group("M5(2).1C4"); // GroupNames label
G:=SmallGroup(128,893);
// by ID
G=gap.SmallGroup(128,893);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,288,723,1123,360,172,4037,124]);
// 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,c*b*c^-1=a^8*b>;
// generators/relations