direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C42.3C4, C42.3C12, C4⋊Q8.3C6, (C4×C12).6C4, (C6×Q8).6C4, (C2×C12).20D4, C4.10D4.C6, (C2×Q8).4C12, C6.37(C23⋊C4), (C6×Q8).156C22, (C2×C4).4(C3×D4), (C2×C4).4(C2×C12), (C3×C4⋊Q8).18C2, (C2×Q8).3(C2×C6), (C2×C12).15(C2×C4), C2.11(C3×C23⋊C4), (C2×C6).78(C22⋊C4), (C3×C4.10D4).2C2, C22.15(C3×C22⋊C4), SmallGroup(192,162)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42.3C4
G = < a,b,c,d | a3=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b2c >
Subgroups: 114 in 60 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C12, C2×C6, C42, C4⋊C4, M4(2), C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×Q8, C4.10D4, C4⋊Q8, C4×C12, C3×C4⋊C4, C3×M4(2), C6×Q8, C42.3C4, C3×C4.10D4, C3×C4⋊Q8, C3×C42.3C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C12, C3×D4, C23⋊C4, C3×C22⋊C4, C42.3C4, C3×C23⋊C4, C3×C42.3C4
►On 48 points
(1 16 39)(2 9 40)(3 10 33)(4 11 34)(5 12 35)(6 13 36)(7 14 37)(8 15 38)(17 29 41)(18 30 42)(19 31 43)(20 32 44)(21 25 45)(22 26 46)(23 27 47)(24 28 48)
(2 24 6 20)(4 22 8 18)(9 28 13 32)(11 26 15 30)(34 46 38 42)(36 44 40 48)
(1 23 5 19)(2 24 6 20)(3 21 7 17)(4 22 8 18)(9 28 13 32)(10 25 14 29)(11 26 15 30)(12 31 16 27)(33 45 37 41)(34 46 38 42)(35 43 39 47)(36 44 40 48)
(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)| (1,16,39)(2,9,40)(3,10,33)(4,11,34)(5,12,35)(6,13,36)(7,14,37)(8,15,38)(17,29,41)(18,30,42)(19,31,43)(20,32,44)(21,25,45)(22,26,46)(23,27,47)(24,28,48), (2,24,6,20)(4,22,8,18)(9,28,13,32)(11,26,15,30)(34,46,38,42)(36,44,40,48), (1,23,5,19)(2,24,6,20)(3,21,7,17)(4,22,8,18)(9,28,13,32)(10,25,14,29)(11,26,15,30)(12,31,16,27)(33,45,37,41)(34,46,38,42)(35,43,39,47)(36,44,40,48), (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( (1,16,39)(2,9,40)(3,10,33)(4,11,34)(5,12,35)(6,13,36)(7,14,37)(8,15,38)(17,29,41)(18,30,42)(19,31,43)(20,32,44)(21,25,45)(22,26,46)(23,27,47)(24,28,48), (2,24,6,20)(4,22,8,18)(9,28,13,32)(11,26,15,30)(34,46,38,42)(36,44,40,48), (1,23,5,19)(2,24,6,20)(3,21,7,17)(4,22,8,18)(9,28,13,32)(10,25,14,29)(11,26,15,30)(12,31,16,27)(33,45,37,41)(34,46,38,42)(35,43,39,47)(36,44,40,48), (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([[(1,16,39),(2,9,40),(3,10,33),(4,11,34),(5,12,35),(6,13,36),(7,14,37),(8,15,38),(17,29,41),(18,30,42),(19,31,43),(20,32,44),(21,25,45),(22,26,46),(23,27,47),(24,28,48)], [(2,24,6,20),(4,22,8,18),(9,28,13,32),(11,26,15,30),(34,46,38,42),(36,44,40,48)], [(1,23,5,19),(2,24,6,20),(3,21,7,17),(4,22,8,18),(9,28,13,32),(10,25,14,29),(11,26,15,30),(12,31,16,27),(33,45,37,41),(34,46,38,42),(35,43,39,47),(36,44,40,48)], [(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)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | ··· | 4E | 4F | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | ··· | 12J | 12K | 12L | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | 1 | 4 | ··· | 4 | 8 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | C3×D4 | C23⋊C4 | C42.3C4 | C3×C23⋊C4 | C3×C42.3C4 |
| kernel | C3×C42.3C4 | C3×C4.10D4 | C3×C4⋊Q8 | C42.3C4 | C4×C12 | C6×Q8 | C4.10D4 | C4⋊Q8 | C42 | C2×Q8 | C2×C12 | C2×C4 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 1 | 2 | 2 | 4 |
Matrix representation of C3×C42.3C4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 3 | 0 | 2 |
| 3 | 2 | 3 | 4 |
| 5 | 5 | 4 | 2 |
| 0 | 4 | 2 | 3 |
| 5 | 0 | 3 | 0 |
| 1 | 5 | 5 | 4 |
| 3 | 0 | 2 | 0 |
| 6 | 4 | 1 | 2 |
| 2 | 6 | 3 | 0 |
| 2 | 1 | 4 | 5 |
| 2 | 3 | 2 | 3 |
| 3 | 4 | 1 | 2 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,3,5,0,3,2,5,4,0,3,4,2,2,4,2,3],[5,1,3,6,0,5,0,4,3,5,2,1,0,4,0,2],[2,2,2,3,6,1,3,4,3,4,2,1,0,5,3,2] >;
C3×C42.3C4 in GAP, Magma, Sage, TeX
C_3\times C_4^2._3C_4
% in TeX
G:=Group("C3xC4^2.3C4"); // GroupNames label
G:=SmallGroup(192,162);
// by ID
G=gap.SmallGroup(192,162);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,680,1683,1522,248,2951,375,172,6053]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=1,d^4=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^2*c>;
// generators/relations