direct product, abelian, monomial, 3-elementary
Aliases: C32×C6, SmallGroup(54,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32×C6 |
| C1 — C32×C6 |
| C1 — C32×C6 |
Generators and relations for C32×C6
G = < a,b,c | a3=b3=c6=1, ab=ba, ac=ca, bc=cb >
Subgroups: 56, all normal (4 characteristic)
C1, C2, C3, C6, C32, C3×C6, C33, C32×C6
Quotients: C1, C2, C3, C6, C32, C3×C6, C33, C32×C6
►Regular action on 54 points
Generators in S54
(1 49 11)(2 50 12)(3 51 7)(4 52 8)(5 53 9)(6 54 10)(13 38 48)(14 39 43)(15 40 44)(16 41 45)(17 42 46)(18 37 47)(19 29 33)(20 30 34)(21 25 35)(22 26 36)(23 27 31)(24 28 32)
(1 20 14)(2 21 15)(3 22 16)(4 23 17)(5 24 18)(6 19 13)(7 36 45)(8 31 46)(9 32 47)(10 33 48)(11 34 43)(12 35 44)(25 40 50)(26 41 51)(27 42 52)(28 37 53)(29 38 54)(30 39 49)
(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)| (1,49,11)(2,50,12)(3,51,7)(4,52,8)(5,53,9)(6,54,10)(13,38,48)(14,39,43)(15,40,44)(16,41,45)(17,42,46)(18,37,47)(19,29,33)(20,30,34)(21,25,35)(22,26,36)(23,27,31)(24,28,32), (1,20,14)(2,21,15)(3,22,16)(4,23,17)(5,24,18)(6,19,13)(7,36,45)(8,31,46)(9,32,47)(10,33,48)(11,34,43)(12,35,44)(25,40,50)(26,41,51)(27,42,52)(28,37,53)(29,38,54)(30,39,49), (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( (1,49,11)(2,50,12)(3,51,7)(4,52,8)(5,53,9)(6,54,10)(13,38,48)(14,39,43)(15,40,44)(16,41,45)(17,42,46)(18,37,47)(19,29,33)(20,30,34)(21,25,35)(22,26,36)(23,27,31)(24,28,32), (1,20,14)(2,21,15)(3,22,16)(4,23,17)(5,24,18)(6,19,13)(7,36,45)(8,31,46)(9,32,47)(10,33,48)(11,34,43)(12,35,44)(25,40,50)(26,41,51)(27,42,52)(28,37,53)(29,38,54)(30,39,49), (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([[(1,49,11),(2,50,12),(3,51,7),(4,52,8),(5,53,9),(6,54,10),(13,38,48),(14,39,43),(15,40,44),(16,41,45),(17,42,46),(18,37,47),(19,29,33),(20,30,34),(21,25,35),(22,26,36),(23,27,31),(24,28,32)], [(1,20,14),(2,21,15),(3,22,16),(4,23,17),(5,24,18),(6,19,13),(7,36,45),(8,31,46),(9,32,47),(10,33,48),(11,34,43),(12,35,44),(25,40,50),(26,41,51),(27,42,52),(28,37,53),(29,38,54),(30,39,49)], [(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)]])
C32×C6 is a maximal subgroup of
C33⋊5C4
54 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3Z | 6A | ··· | 6Z |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C3 | C6 |
| kernel | C32×C6 | C33 | C3×C6 | C32 |
| # reps | 1 | 1 | 26 | 26 |
Matrix representation of C32×C6 ►in GL3(𝔽7) generated by
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 4 |
| 4 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 4 |
| 2 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
G:=sub<GL(3,GF(7))| [2,0,0,0,2,0,0,0,4],[4,0,0,0,1,0,0,0,4],[2,0,0,0,3,0,0,0,3] >;
C32×C6 in GAP, Magma, Sage, TeX
C_3^2\times C_6
% in TeX
G:=Group("C3^2xC6"); // GroupNames label
G:=SmallGroup(54,15);
// by ID
G=gap.SmallGroup(54,15);
# by ID
G:=PCGroup([4,-2,-3,-3,-3]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^6=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations