direct product, metabelian, soluble, monomial, A-group
Aliases: C12×C32⋊C4, C33⋊2C42, (C3×C12)⋊4C12, C3⋊Dic3⋊6C12, C32⋊2(C4×C12), (C32×C12)⋊2C4, C2.2(C6×C32⋊C4), (C4×C3⋊S3).11C6, C3⋊S3.5(C2×C12), (C3×C3⋊Dic3)⋊2C4, (C2×C32⋊C4).6C6, C6.21(C2×C32⋊C4), (C12×C3⋊S3).13C2, (C3×C6).12(C2×C12), (C6×C32⋊C4).13C2, (C6×C3⋊S3).38C22, (C32×C6).10(C2×C4), (C2×C3⋊S3).13(C2×C6), (C3×C3⋊S3).15(C2×C4), SmallGroup(432,630)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C12×C32⋊C4 |
Generators and relations for C12×C32⋊C4
G = < a,b,c,d | a12=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Subgroups: 444 in 104 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C42, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C32⋊C4, S3×C6, C2×C3⋊S3, C4×C12, C3×C3⋊S3, C32×C6, S3×C12, C4×C3⋊S3, C2×C32⋊C4, C3×C3⋊Dic3, C32×C12, C3×C32⋊C4, C6×C3⋊S3, C4×C32⋊C4, C12×C3⋊S3, C6×C32⋊C4, C12×C32⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, C2×C12, C32⋊C4, C4×C12, C2×C32⋊C4, C3×C32⋊C4, C4×C32⋊C4, C6×C32⋊C4, C12×C32⋊C4
►On 48 points
(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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 33 20 37)(2 34 21 38)(3 35 22 39)(4 36 23 40)(5 25 24 41)(6 26 13 42)(7 27 14 43)(8 28 15 44)(9 29 16 45)(10 30 17 46)(11 31 18 47)(12 32 19 48)
G:=sub<Sym(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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,33,20,37)(2,34,21,38)(3,35,22,39)(4,36,23,40)(5,25,24,41)(6,26,13,42)(7,27,14,43)(8,28,15,44)(9,29,16,45)(10,30,17,46)(11,31,18,47)(12,32,19,48)>;
G:=Group( (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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,33,20,37)(2,34,21,38)(3,35,22,39)(4,36,23,40)(5,25,24,41)(6,26,13,42)(7,27,14,43)(8,28,15,44)(9,29,16,45)(10,30,17,46)(11,31,18,47)(12,32,19,48) );
G=PermutationGroup([[(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)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,33,20,37),(2,34,21,38),(3,35,22,39),(4,36,23,40),(5,25,24,41),(6,26,13,42),(7,27,14,43),(8,28,15,44),(9,29,16,45),(10,30,17,46),(11,31,18,47),(12,32,19,48)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | ··· | 4L | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 12Q | ··· | 12AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 9 | ··· | 9 | 1 | 1 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 9 | ··· | 9 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C12 | C12 | C12 | C32⋊C4 | C2×C32⋊C4 | C3×C32⋊C4 | C4×C32⋊C4 | C6×C32⋊C4 | C12×C32⋊C4 |
| kernel | C12×C32⋊C4 | C12×C3⋊S3 | C6×C32⋊C4 | C4×C32⋊C4 | C3×C3⋊Dic3 | C32×C12 | C3×C32⋊C4 | C4×C3⋊S3 | C2×C32⋊C4 | C3⋊Dic3 | C3×C12 | C32⋊C4 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 2 | 4 | 4 | 4 | 16 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C12×C32⋊C4 ►in GL4(𝔽13) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 9 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 0 | 5 | 0 | 0 |
| 5 | 0 | 0 | 0 |
G:=sub<GL(4,GF(13))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[9,0,0,0,0,3,0,0,0,0,9,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,9],[0,0,0,5,0,0,5,0,5,0,0,0,0,5,0,0] >;
C12×C32⋊C4 in GAP, Magma, Sage, TeX
C_{12}\times C_3^2\rtimes C_4 % in TeX
G:=Group("C12xC3^2:C4"); // GroupNames label
G:=SmallGroup(432,630);
// by ID
G=gap.SmallGroup(432,630);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,176,14117,362,18822,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations