p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).1C8, C23.13M4(2), M5(2).23C22, C8.5(C2×C8), C8.39(C4⋊C4), C4.20(C4⋊C8), (C2×C8).33Q8, C8.27(C2×Q8), C8⋊C4.7C4, (C2×C8).393D4, C8.137(C2×D4), C8.C8⋊10C2, C4.29(C22×C8), C22.14(C4⋊C8), C42.170(C2×C4), (C4×C8).157C22, (C2×C8).606C23, (C2×C4).26M4(2), C42⋊C2.22C4, C8○2M4(2).6C2, (C2×M5(2)).26C2, (C2×M4(2)).14C4, (C22×C8).418C22, C22.24(C2×M4(2)), C2.16(C2×C4⋊C8), C4.82(C2×C4⋊C4), (C2×C8).88(C2×C4), (C2×C4).27(C2×C8), (C2×C4).140(C4⋊C4), (C2×C4).561(C22×C4), (C22×C4).288(C2×C4), SmallGroup(128,885)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).1C8
G = < a,b,c | a8=b2=1, c8=a4, bab=a5, cac-1=a-1, cbc-1=a4b >
Subgroups: 100 in 76 conjugacy classes, 58 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C2×C16, M5(2), M5(2), C42⋊C2, C22×C8, C2×M4(2), C8.C8, C8○2M4(2), C2×M5(2), M4(2).1C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C2×C4⋊C8, M4(2).1C8
►On 32 points
(1 27 13 23 9 19 5 31)(2 32 6 20 10 24 14 28)(3 29 15 25 11 21 7 17)(4 18 8 22 12 26 16 30)
(1 9)(3 11)(5 13)(7 15)(18 26)(20 28)(22 30)(24 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)
G:=sub<Sym(32)| (1,27,13,23,9,19,5,31)(2,32,6,20,10,24,14,28)(3,29,15,25,11,21,7,17)(4,18,8,22,12,26,16,30), (1,9)(3,11)(5,13)(7,15)(18,26)(20,28)(22,30)(24,32), (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)>;
G:=Group( (1,27,13,23,9,19,5,31)(2,32,6,20,10,24,14,28)(3,29,15,25,11,21,7,17)(4,18,8,22,12,26,16,30), (1,9)(3,11)(5,13)(7,15)(18,26)(20,28)(22,30)(24,32), (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) );
G=PermutationGroup([[(1,27,13,23,9,19,5,31),(2,32,6,20,10,24,14,28),(3,29,15,25,11,21,7,17),(4,18,8,22,12,26,16,30)], [(1,9),(3,11),(5,13),(7,15),(18,26),(20,28),(22,30),(24,32)], [(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)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | Q8 | M4(2) | M4(2) | M4(2).1C8 |
| kernel | M4(2).1C8 | C8.C8 | C8○2M4(2) | C2×M5(2) | C8⋊C4 | C42⋊C2 | C2×M4(2) | M4(2) | C2×C8 | C2×C8 | C2×C4 | C23 | C1 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 16 | 2 | 2 | 2 | 2 | 4 |
Matrix representation of M4(2).1C8 ►in GL4(𝔽17) generated by
| 0 | 4 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 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 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
G:=sub<GL(4,GF(17))| [0,1,0,0,4,0,0,0,0,0,0,13,0,0,1,0],[16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16],[0,0,8,0,0,0,0,8,1,0,0,0,0,1,0,0] >;
M4(2).1C8 in GAP, Magma, Sage, TeX
M_4(2)._1C_8
% in TeX
G:=Group("M4(2).1C8"); // GroupNames label
G:=SmallGroup(128,885);
// by ID
G=gap.SmallGroup(128,885);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,64,723,2019,248,102,124]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=1,c^8=a^4,b*a*b=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^4*b>;
// generators/relations