direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×M4(2), C12○M4(2), C24⋊14C22, C23.4C12, C12.53C23, (C2×C8)⋊6C6, C8⋊4(C2×C6), (C2×C24)⋊14C2, (C2×C4).6C12, C4○(C3×M4(2)), (C2×C12).15C4, C4.10(C2×C12), C12.47(C2×C4), C12○(C3×M4(2)), (C22×C6).4C4, (C22×C4).8C6, C4.11(C22×C6), C22.6(C2×C12), C2.6(C22×C12), C6.34(C22×C4), (C22×C12).16C2, (C2×C12).127C22, (C2×C6).23(C2×C4), (C2×C4).33(C2×C6), SmallGroup(96,177)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×M4(2)
G = < a,b,c | a6=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Subgroups: 76 in 68 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C24, C2×C12, C2×C12, C22×C6, C2×M4(2), C2×C24, C3×M4(2), C22×C12, C6×M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, M4(2), C22×C4, C2×C12, C22×C6, C2×M4(2), C3×M4(2), C22×C12, C6×M4(2)
►On 48 points
(1 11 34 18 45 30)(2 12 35 19 46 31)(3 13 36 20 47 32)(4 14 37 21 48 25)(5 15 38 22 41 26)(6 16 39 23 42 27)(7 9 40 24 43 28)(8 10 33 17 44 29)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)
G:=sub<Sym(48)| (1,11,34,18,45,30)(2,12,35,19,46,31)(3,13,36,20,47,32)(4,14,37,21,48,25)(5,15,38,22,41,26)(6,16,39,23,42,27)(7,9,40,24,43,28)(8,10,33,17,44,29), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)>;
G:=Group( (1,11,34,18,45,30)(2,12,35,19,46,31)(3,13,36,20,47,32)(4,14,37,21,48,25)(5,15,38,22,41,26)(6,16,39,23,42,27)(7,9,40,24,43,28)(8,10,33,17,44,29), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48) );
G=PermutationGroup([[(1,11,34,18,45,30),(2,12,35,19,46,31),(3,13,36,20,47,32),(4,14,37,21,48,25),(5,15,38,22,41,26),(6,16,39,23,42,27),(7,9,40,24,43,28),(8,10,33,17,44,29)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(42,46),(44,48)]])
C6×M4(2) is a maximal subgroup of
C24.D4 M4(2)⋊Dic3 C12.3C42 (C2×C24)⋊C4 C12.20C42 C12.4C42 M4(2)⋊4Dic3 C12.21C42 Dic3⋊4M4(2) C12.88(C2×Q8) C23.51D12 C23.52D12 C12.7C42 C23.8Dic6 C23.9Dic6 D6⋊6M4(2) C24⋊D4 C24⋊21D4 D6⋊C8⋊40C2 C23.53D12 M4(2).31D6 C23.54D12 C24⋊2D4 C24⋊3D4 C24.4D4 M4(2)⋊24D6 M4(2)⋊26D6 C24.9C23
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | M4(2) | C3×M4(2) |
| kernel | C6×M4(2) | C2×C24 | C3×M4(2) | C22×C12 | C2×M4(2) | C2×C12 | C22×C6 | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 4 | 1 | 2 | 6 | 2 | 4 | 8 | 2 | 12 | 4 | 4 | 8 |
Matrix representation of C6×M4(2) ►in GL3(𝔽73) generated by
| 9 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 72 | 0 | 0 |
| 0 | 0 | 72 |
| 0 | 27 | 0 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 72 |
G:=sub<GL(3,GF(73))| [9,0,0,0,1,0,0,0,1],[72,0,0,0,0,27,0,72,0],[72,0,0,0,1,0,0,0,72] >;
C6×M4(2) in GAP, Magma, Sage, TeX
C_6\times M_4(2)
% in TeX
G:=Group("C6xM4(2)"); // GroupNames label
G:=SmallGroup(96,177);
// by ID
G=gap.SmallGroup(96,177);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,144,601,88]);
// Polycyclic
G:=Group<a,b,c|a^6=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations