Aliases: C62.2C9, C9.A4⋊2C3, C9.5(C3×A4), (C3×C9).3A4, C9.(C3.A4), (C2×C18).3C9, (C6×C18).2C3, C22⋊2(C27⋊C3), C32.(C3.A4), (C2×C18).5C32, (C2×C6).6(C3×C9), C3.3(C3×C3.A4), SmallGroup(324,45)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.C9
G = < a,b,c | a6=b6=1, c9=b2, ab=ba, cac-1=ab-1, cbc-1=a3b4 >
Smallest permutation representation of C62.C9
►On 54 points
Generators in S54
(2 42 20 33 11 51)(3 52 12 34 21 43)(5 45 23 36 14 54)(6 28 15 37 24 46)(8 48 26 39 17 30)(9 31 18 40 27 49)
(1 50 10 32 19 41)(2 20 11)(3 52 12 34 21 43)(4 53 13 35 22 44)(5 23 14)(6 28 15 37 24 46)(7 29 16 38 25 47)(8 26 17)(9 31 18 40 27 49)(30 48 39)(33 51 42)(36 54 45)
(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)
G:=sub<Sym(54)| (2,42,20,33,11,51)(3,52,12,34,21,43)(5,45,23,36,14,54)(6,28,15,37,24,46)(8,48,26,39,17,30)(9,31,18,40,27,49), (1,50,10,32,19,41)(2,20,11)(3,52,12,34,21,43)(4,53,13,35,22,44)(5,23,14)(6,28,15,37,24,46)(7,29,16,38,25,47)(8,26,17)(9,31,18,40,27,49)(30,48,39)(33,51,42)(36,54,45), (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)>;
G:=Group( (2,42,20,33,11,51)(3,52,12,34,21,43)(5,45,23,36,14,54)(6,28,15,37,24,46)(8,48,26,39,17,30)(9,31,18,40,27,49), (1,50,10,32,19,41)(2,20,11)(3,52,12,34,21,43)(4,53,13,35,22,44)(5,23,14)(6,28,15,37,24,46)(7,29,16,38,25,47)(8,26,17)(9,31,18,40,27,49)(30,48,39)(33,51,42)(36,54,45), (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) );
G=PermutationGroup([[(2,42,20,33,11,51),(3,52,12,34,21,43),(5,45,23,36,14,54),(6,28,15,37,24,46),(8,48,26,39,17,30),(9,31,18,40,27,49)], [(1,50,10,32,19,41),(2,20,11),(3,52,12,34,21,43),(4,53,13,35,22,44),(5,23,14),(6,28,15,37,24,46),(7,29,16,38,25,47),(8,26,17),(9,31,18,40,27,49),(30,48,39),(33,51,42),(36,54,45)], [(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)]])
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | ··· | 6H | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 18A | ··· | 18R | 27A | ··· | 27R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 27 | ··· | 27 |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 12 | ··· | 12 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||||
| image | C1 | C3 | C3 | C9 | C9 | A4 | C3.A4 | C3×A4 | C3.A4 | C27⋊C3 | C62.C9 |
| kernel | C62.C9 | C9.A4 | C6×C18 | C2×C18 | C62 | C3×C9 | C9 | C9 | C32 | C22 | C1 |
| # reps | 1 | 6 | 2 | 12 | 6 | 1 | 4 | 2 | 2 | 6 | 18 |
Matrix representation of C62.C9 ►in GL3(𝔽109) generated by
| 1 | 0 | 0 |
| 101 | 64 | 0 |
| 90 | 0 | 46 |
| 64 | 0 | 0 |
| 0 | 64 | 0 |
| 71 | 0 | 45 |
| 66 | 107 | 0 |
| 0 | 43 | 1 |
| 53 | 4 | 0 |
G:=sub<GL(3,GF(109))| [1,101,90,0,64,0,0,0,46],[64,0,71,0,64,0,0,0,45],[66,0,53,107,43,4,0,1,0] >;
C62.C9 in GAP, Magma, Sage, TeX
C_6^2.C_9
% in TeX
G:=Group("C6^2.C9"); // GroupNames label
G:=SmallGroup(324,45);
// by ID
G=gap.SmallGroup(324,45);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,54,361,68,4864,8753]);
// Polycyclic
G:=Group<a,b,c|a^6=b^6=1,c^9=b^2,a*b=b*a,c*a*c^-1=a*b^-1,c*b*c^-1=a^3*b^4>;
// generators/relations
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