direct product, abelian, monomial
Aliases: C6×C12, SmallGroup(72,36)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C6×C12 |
| C1 — C6×C12 |
| C1 — C6×C12 |
Generators and relations for C6×C12
G = < a,b | a6=b12=1, ab=ba >
Smallest permutation representation of C6×C12
►Regular action on 72 points
Generators in S72
(1 65 32 42 17 51)(2 66 33 43 18 52)(3 67 34 44 19 53)(4 68 35 45 20 54)(5 69 36 46 21 55)(6 70 25 47 22 56)(7 71 26 48 23 57)(8 72 27 37 24 58)(9 61 28 38 13 59)(10 62 29 39 14 60)(11 63 30 40 15 49)(12 64 31 41 16 50)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
G:=sub<Sym(72)| (1,65,32,42,17,51)(2,66,33,43,18,52)(3,67,34,44,19,53)(4,68,35,45,20,54)(5,69,36,46,21,55)(6,70,25,47,22,56)(7,71,26,48,23,57)(8,72,27,37,24,58)(9,61,28,38,13,59)(10,62,29,39,14,60)(11,63,30,40,15,49)(12,64,31,41,16,50), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)>;
G:=Group( (1,65,32,42,17,51)(2,66,33,43,18,52)(3,67,34,44,19,53)(4,68,35,45,20,54)(5,69,36,46,21,55)(6,70,25,47,22,56)(7,71,26,48,23,57)(8,72,27,37,24,58)(9,61,28,38,13,59)(10,62,29,39,14,60)(11,63,30,40,15,49)(12,64,31,41,16,50), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72) );
G=PermutationGroup([[(1,65,32,42,17,51),(2,66,33,43,18,52),(3,67,34,44,19,53),(4,68,35,45,20,54),(5,69,36,46,21,55),(6,70,25,47,22,56),(7,71,26,48,23,57),(8,72,27,37,24,58),(9,61,28,38,13,59),(10,62,29,39,14,60),(11,63,30,40,15,49),(12,64,31,41,16,50)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)]])
C6×C12 is a maximal subgroup of
C12.58D6 C6.Dic6 C12⋊Dic3 C6.11D12 C12.59D6
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 6A | ··· | 6X | 12A | ··· | 12AF |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 |
| kernel | C6×C12 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 8 | 32 |
Matrix representation of C6×C12 ►in GL2(𝔽13) generated by
| 1 | 0 |
| 0 | 10 |
| 6 | 0 |
| 0 | 12 |
G:=sub<GL(2,GF(13))| [1,0,0,10],[6,0,0,12] >;
C6×C12 in GAP, Magma, Sage, TeX
C_6\times C_{12} % in TeX
G:=Group("C6xC12"); // GroupNames label
G:=SmallGroup(72,36);
// by ID
G=gap.SmallGroup(72,36);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-2,180]);
// Polycyclic
G:=Group<a,b|a^6=b^12=1,a*b=b*a>;
// generators/relations
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