direct product, metabelian, soluble, monomial, A-group
Aliases: C12×A4, (C2×C6)⋊2C12, (C22×C12)⋊C3, C22⋊(C3×C12), C6.8(C2×A4), C2.1(C6×A4), C23.(C3×C6), (C22×C4)⋊C32, (C6×A4).4C2, (C2×A4).2C6, (C22×C6).4C6, SmallGroup(144,155)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C12×A4 |
Generators and relations for C12×A4
G = < a,b,c,d | a12=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of C12×A4
►On 36 points
Generators in S36
(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)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 24 26)(2 13 27)(3 14 28)(4 15 29)(5 16 30)(6 17 31)(7 18 32)(8 19 33)(9 20 34)(10 21 35)(11 22 36)(12 23 25)
G:=sub<Sym(36)| (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), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,24,26)(2,13,27)(3,14,28)(4,15,29)(5,16,30)(6,17,31)(7,18,32)(8,19,33)(9,20,34)(10,21,35)(11,22,36)(12,23,25)>;
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), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,24,26)(2,13,27)(3,14,28)(4,15,29)(5,16,30)(6,17,31)(7,18,32)(8,19,33)(9,20,34)(10,21,35)(11,22,36)(12,23,25) );
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)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,24,26),(2,13,27),(3,14,28),(4,15,29),(5,16,30),(6,17,31),(7,18,32),(8,19,33),(9,20,34),(10,21,35),(11,22,36),(12,23,25)]])
C12×A4 is a maximal subgroup of
C12.12S4 A4⋊Dic6 C12⋊S4
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | ··· | 6L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12T |
| 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 | 12 | ··· | 12 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | |||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C12 | C12 | A4 | C2×A4 | C3×A4 | C4×A4 | C6×A4 | C12×A4 |
| kernel | C12×A4 | C6×A4 | C4×A4 | C22×C12 | C3×A4 | C2×A4 | C22×C6 | A4 | C2×C6 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 12 | 4 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C12×A4 ►in GL3(𝔽13) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 1 |
| 12 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 12 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 9 | 0 | 0 |
G:=sub<GL(3,GF(13))| [6,0,0,0,6,0,0,0,6],[12,0,0,0,12,0,0,0,1],[12,0,0,0,1,0,0,0,12],[0,0,9,4,0,0,0,4,0] >;
C12×A4 in GAP, Magma, Sage, TeX
C_{12}\times A_4 % in TeX
G:=Group("C12xA4"); // GroupNames label
G:=SmallGroup(144,155);
// by ID
G=gap.SmallGroup(144,155);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-2,2,108,1090,1955]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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