direct product, abelian, monomial, 3-elementary
Aliases: C3×C12, SmallGroup(36,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C12 |
| C1 — C3×C12 |
| C1 — C3×C12 |
Generators and relations for C3×C12
G = < a,b | a3=b12=1, ab=ba >
Smallest permutation representation of C3×C12
►Regular action on 36 points
Generators in S36
(1 16 27)(2 17 28)(3 18 29)(4 19 30)(5 20 31)(6 21 32)(7 22 33)(8 23 34)(9 24 35)(10 13 36)(11 14 25)(12 15 26)
(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)
G:=sub<Sym(36)| (1,16,27)(2,17,28)(3,18,29)(4,19,30)(5,20,31)(6,21,32)(7,22,33)(8,23,34)(9,24,35)(10,13,36)(11,14,25)(12,15,26), (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)>;
G:=Group( (1,16,27)(2,17,28)(3,18,29)(4,19,30)(5,20,31)(6,21,32)(7,22,33)(8,23,34)(9,24,35)(10,13,36)(11,14,25)(12,15,26), (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) );
G=PermutationGroup([[(1,16,27),(2,17,28),(3,18,29),(4,19,30),(5,20,31),(6,21,32),(7,22,33),(8,23,34),(9,24,35),(10,13,36),(11,14,25),(12,15,26)], [(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)]])
C3×C12 is a maximal subgroup of
C32⋊4C8 C32⋊4Q8 C12⋊S3
36 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6H | 12A | ··· | 12P |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C4 | C6 | C12 |
| kernel | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 16 |
Matrix representation of C3×C12 ►in GL2(𝔽13) generated by
| 9 | 0 |
| 0 | 9 |
| 7 | 0 |
| 0 | 2 |
G:=sub<GL(2,GF(13))| [9,0,0,9],[7,0,0,2] >;
C3×C12 in GAP, Magma, Sage, TeX
C_3\times C_{12} % in TeX
G:=Group("C3xC12"); // GroupNames label
G:=SmallGroup(36,8);
// by ID
G=gap.SmallGroup(36,8);
# by ID
G:=PCGroup([4,-2,-3,-3,-2,72]);
// Polycyclic
G:=Group<a,b|a^3=b^12=1,a*b=b*a>;
// generators/relations
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