direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4.10D4, C12.59D4, M4(2).1C6, (C2×C4).C12, (C2×C12).2C4, C4.10(C3×D4), (C6×Q8).6C2, (C2×Q8).3C6, C22.4(C2×C12), C6.23(C22⋊C4), (C2×C12).60C22, (C3×M4(2)).3C2, (C2×C4).2(C2×C6), (C2×C6).21(C2×C4), C2.5(C3×C22⋊C4), SmallGroup(96,51)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.10D4
G = < a,b,c,d | a3=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >
Smallest permutation representation of C3×C4.10D4
►On 48 points
Generators in S48
(1 12 23)(2 13 24)(3 14 17)(4 15 18)(5 16 19)(6 9 20)(7 10 21)(8 11 22)(25 35 45)(26 36 46)(27 37 47)(28 38 48)(29 39 41)(30 40 42)(31 33 43)(32 34 44)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)
(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)
(1 44 7 46 5 48 3 42)(2 45 4 43 6 41 8 47)(9 29 11 27 13 25 15 31)(10 26 16 28 14 30 12 32)(17 40 23 34 21 36 19 38)(18 33 20 39 22 37 24 35)
G:=sub<Sym(48)| (1,12,23)(2,13,24)(3,14,17)(4,15,18)(5,16,19)(6,9,20)(7,10,21)(8,11,22)(25,35,45)(26,36,46)(27,37,47)(28,38,48)(29,39,41)(30,40,42)(31,33,43)(32,34,44), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44), (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), (1,44,7,46,5,48,3,42)(2,45,4,43,6,41,8,47)(9,29,11,27,13,25,15,31)(10,26,16,28,14,30,12,32)(17,40,23,34,21,36,19,38)(18,33,20,39,22,37,24,35)>;
G:=Group( (1,12,23)(2,13,24)(3,14,17)(4,15,18)(5,16,19)(6,9,20)(7,10,21)(8,11,22)(25,35,45)(26,36,46)(27,37,47)(28,38,48)(29,39,41)(30,40,42)(31,33,43)(32,34,44), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44), (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), (1,44,7,46,5,48,3,42)(2,45,4,43,6,41,8,47)(9,29,11,27,13,25,15,31)(10,26,16,28,14,30,12,32)(17,40,23,34,21,36,19,38)(18,33,20,39,22,37,24,35) );
G=PermutationGroup([[(1,12,23),(2,13,24),(3,14,17),(4,15,18),(5,16,19),(6,9,20),(7,10,21),(8,11,22),(25,35,45),(26,36,46),(27,37,47),(28,38,48),(29,39,41),(30,40,42),(31,33,43),(32,34,44)], [(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16),(17,19,21,23),(18,24,22,20),(25,27,29,31),(26,32,30,28),(33,35,37,39),(34,40,38,36),(41,43,45,47),(42,48,46,44)], [(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)], [(1,44,7,46,5,48,3,42),(2,45,4,43,6,41,8,47),(9,29,11,27,13,25,15,31),(10,26,16,28,14,30,12,32),(17,40,23,34,21,36,19,38),(18,33,20,39,22,37,24,35)]])
C3×C4.10D4 is a maximal subgroup of
(C2×C4).D12 (C2×C12).D4 M4(2).21D6 D12.4D4 D12.5D4 D12.6D4 D12.7D4
33 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | C3×D4 | C4.10D4 | C3×C4.10D4 |
| kernel | C3×C4.10D4 | C3×M4(2) | C6×Q8 | C4.10D4 | C2×C12 | M4(2) | C2×Q8 | C2×C4 | C12 | C4 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 2 | 4 | 1 | 2 |
Matrix representation of C3×C4.10D4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 3 | 1 | 4 |
| 6 | 3 | 5 | 3 |
| 2 | 2 | 1 | 0 |
| 0 | 6 | 3 | 3 |
| 1 | 1 | 5 | 0 |
| 2 | 5 | 5 | 1 |
| 4 | 5 | 1 | 4 |
| 4 | 5 | 2 | 0 |
| 3 | 0 | 5 | 1 |
| 5 | 2 | 0 | 4 |
| 5 | 5 | 6 | 5 |
| 1 | 0 | 3 | 3 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,6,2,0,3,3,2,6,1,5,1,3,4,3,0,3],[1,2,4,4,1,5,5,5,5,5,1,2,0,1,4,0],[3,5,5,1,0,2,5,0,5,0,6,3,1,4,5,3] >;
C3×C4.10D4 in GAP, Magma, Sage, TeX
C_3\times C_4._{10}D_4 % in TeX
G:=Group("C3xC4.10D4"); // GroupNames label
G:=SmallGroup(96,51);
// by ID
G=gap.SmallGroup(96,51);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,295,1443,1090,88]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
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