p-group, metabelian, nilpotent (class 3), monomial
Aliases: M5(2)⋊6C4, M4(2).2C8, (C2×C16)⋊4C4, C8.35(C4⋊C4), C4.16(C4⋊C8), (C2×C8).28Q8, (C2×C8).375D4, C22⋊C4.2C8, C22.8(C4×C8), C23.6(C2×C8), C4.8(C8⋊C4), (C2×C4).55C42, C8.56(C22⋊C4), C4.21(C22⋊C8), C22.10(C4⋊C8), (C2×M5(2)).9C2, (C2×C4).12M4(2), C42⋊C2.13C4, (C2×M4(2)).19C4, C8○2M4(2).12C2, C22.24(C22⋊C8), (C22×C8).369C22, C4.28(C2.C42), C2.16(C22.7C42), (C2×C4).14(C2×C8), (C2×C8).238(C2×C4), (C2×C4).106(C4⋊C4), (C22×C4).166(C2×C4), (C2×C4).348(C22⋊C4), SmallGroup(128,110)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).C8
G = < a,b,c | a8=b2=1, c8=a4, bab=a5, cac-1=ab, bc=cb >
Subgroups: 96 in 66 conjugacy classes, 42 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C4×C8, C8⋊C4, C2×C16, C2×C16, M5(2), M5(2), C42⋊C2, C22×C8, C2×M4(2), C8○2M4(2), C2×M5(2), M4(2).C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C22.7C42, M4(2).C8
►On 32 points
(1 17 5 21 9 25 13 29)(2 8 14 4 10 16 6 12)(3 19 7 23 11 27 15 31)(18 32 30 28 26 24 22 20)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)
(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)| (1,17,5,21,9,25,13,29)(2,8,14,4,10,16,6,12)(3,19,7,23,11,27,15,31)(18,32,30,28,26,24,22,20), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26), (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( (1,17,5,21,9,25,13,29)(2,8,14,4,10,16,6,12)(3,19,7,23,11,27,15,31)(18,32,30,28,26,24,22,20), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26), (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([[(1,17,5,21,9,25,13,29),(2,8,14,4,10,16,6,12),(3,19,7,23,11,27,15,31),(18,32,30,28,26,24,22,20)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26)], [(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 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8 | D4 | Q8 | M4(2) | M4(2).C8 |
| kernel | M4(2).C8 | C8○2M4(2) | C2×M5(2) | C2×C16 | M5(2) | C42⋊C2 | C2×M4(2) | C22⋊C4 | M4(2) | C2×C8 | C2×C8 | C2×C4 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 8 | 8 | 3 | 1 | 4 | 4 |
Matrix representation of M4(2).C8 ►in GL4(𝔽17) generated by
| 0 | 15 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
G:=sub<GL(4,GF(17))| [0,2,0,0,15,0,0,0,0,0,2,0,0,0,0,15],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[0,0,8,0,0,0,0,8,1,0,0,0,0,1,0,0] >;
M4(2).C8 in GAP, Magma, Sage, TeX
M_4(2).C_8
% in TeX
G:=Group("M4(2).C8"); // GroupNames label
G:=SmallGroup(128,110);
// by ID
G=gap.SmallGroup(128,110);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,723,136,2804,124]);
// 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*b,b*c=c*b>;
// generators/relations