metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.196D6, M4(2).23D6, C4≀C2⋊7S3, D4⋊S3⋊4C4, C3⋊2(C8○D8), C3⋊C8.37D4, D4.S3⋊4C4, D4.4(C4×S3), C3⋊Q16⋊4C4, C6.39(C4×D4), Q8.9(C4×S3), D12.C4⋊9C2, Q8⋊2S3⋊4C4, C4○D4.37D6, D12.7(C2×C4), C4.203(S3×D4), C42⋊4S3⋊7C2, C12.362(C2×D4), D4.Dic3⋊2C2, Dic6.7(C2×C4), C12.20(C22×C4), (C4×C12).51C22, Q8.13D6.2C2, C12.53D4⋊10C2, (C2×C12).264C23, C4○D12.13C22, C4.Dic3.9C22, C22.9(D4⋊2S3), C2.23(Dic3⋊4D4), (C3×M4(2)).25C22, (C4×C3⋊C8)⋊3C2, C3⋊C8.8(C2×C4), C4.20(S3×C2×C4), (C3×C4≀C2)⋊12C2, (C3×D4).7(C2×C4), (C3×Q8).7(C2×C4), (C2×C6).35(C4○D4), (C2×C3⋊C8).222C22, (C3×C4○D4).5C22, (C2×C4).370(C22×S3), SmallGroup(192,383)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.196D6
G = < a,b,c,d | a4=b4=c6=1, d2=cbc-1=b-1, ab=ba, cac-1=ab-1, ad=da, bd=db, dcd-1=b-1c-1 >
Subgroups: 240 in 106 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4○D4, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4×C8, C4≀C2, C4≀C2, C8.C4, C8○D4, C4○D8, S3×C8, C8⋊S3, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4×C12, C3×M4(2), C4○D12, C3×C4○D4, C8○D8, C4×C3⋊C8, C42⋊4S3, C12.53D4, C3×C4≀C2, D12.C4, D4.Dic3, Q8.13D6, C42.196D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, S3×C2×C4, S3×D4, D4⋊2S3, C8○D8, Dic3⋊4D4, C42.196D6
►On 48 points
(1 5)(2 6)(3 7)(4 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)
(1 22 39 13 46 26)(2 25 47 12 40 21)(3 20 33 11 48 32)(4 31 41 10 34 19)(5 18 35 9 42 30)(6 29 43 16 36 17)(7 24 37 15 44 28)(8 27 45 14 38 23)
(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)
G:=sub<Sym(48)| (1,5)(2,6)(3,7)(4,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44), (1,22,39,13,46,26)(2,25,47,12,40,21)(3,20,33,11,48,32)(4,31,41,10,34,19)(5,18,35,9,42,30)(6,29,43,16,36,17)(7,24,37,15,44,28)(8,27,45,14,38,23), (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)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,39,37,35)(34,40,38,36)(41,47,45,43)(42,48,46,44), (1,22,39,13,46,26)(2,25,47,12,40,21)(3,20,33,11,48,32)(4,31,41,10,34,19)(5,18,35,9,42,30)(6,29,43,16,36,17)(7,24,37,15,44,28)(8,27,45,14,38,23), (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) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,39,37,35),(34,40,38,36),(41,47,45,43),(42,48,46,44)], [(1,22,39,13,46,26),(2,25,47,12,40,21),(3,20,33,11,48,32),(4,31,41,10,34,19),(5,18,35,9,42,30),(6,29,43,16,36,17),(7,24,37,15,44,28),(8,27,45,14,38,23)], [(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)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | ··· | 8L | 8M | 8N | 12A | 12B | 12C | ··· | 12G | 12H | 24A | 24B |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 12 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | 12 | 2 | 4 | 8 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4×S3 | C4×S3 | C8○D8 | S3×D4 | D4⋊2S3 | C42.196D6 |
| kernel | C42.196D6 | C4×C3⋊C8 | C42⋊4S3 | C12.53D4 | C3×C4≀C2 | D12.C4 | D4.Dic3 | Q8.13D6 | D4⋊S3 | D4.S3 | Q8⋊2S3 | C3⋊Q16 | C4≀C2 | C3⋊C8 | C42 | M4(2) | C4○D4 | C2×C6 | D4 | Q8 | C3 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 1 | 4 |
Matrix representation of C42.196D6 ►in GL4(𝔽5) generated by
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 1 |
| 2 | 1 | 3 | 0 |
| 4 | 0 | 0 | 3 |
| 4 | 0 | 0 | 3 |
| 0 | 4 | 2 | 4 |
| 3 | 4 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 3 | 0 | 2 |
| 0 | 0 | 0 | 1 |
| 0 | 4 | 1 | 2 |
| 3 | 4 | 0 | 0 |
| 3 | 2 | 0 | 3 |
| 4 | 0 | 0 | 4 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [0,0,2,4,0,0,1,0,0,3,3,0,2,1,0,3],[4,0,3,1,0,4,4,0,0,2,1,0,3,4,0,1],[0,1,0,0,1,3,0,4,0,0,0,1,0,2,1,2],[3,3,4,0,4,2,0,0,0,0,0,1,0,3,4,0] >;
C42.196D6 in GAP, Magma, Sage, TeX
C_4^2._{196}D_6 % in TeX
G:=Group("C4^2.196D6"); // GroupNames label
G:=SmallGroup(192,383);
// by ID
G=gap.SmallGroup(192,383);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,555,58,136,1684,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations