direct product, abelian, monomial, 3-elementary
Aliases: C3×C18, SmallGroup(54,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C18 |
| C1 — C3×C18 |
| C1 — C3×C18 |
Generators and relations for C3×C18
G = < a,b | a3=b18=1, ab=ba >
Smallest permutation representation of C3×C18
►Regular action on 54 points
Generators in S54
(1 28 48)(2 29 49)(3 30 50)(4 31 51)(5 32 52)(6 33 53)(7 34 54)(8 35 37)(9 36 38)(10 19 39)(11 20 40)(12 21 41)(13 22 42)(14 23 43)(15 24 44)(16 25 45)(17 26 46)(18 27 47)
(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)| (1,28,48)(2,29,49)(3,30,50)(4,31,51)(5,32,52)(6,33,53)(7,34,54)(8,35,37)(9,36,38)(10,19,39)(11,20,40)(12,21,41)(13,22,42)(14,23,43)(15,24,44)(16,25,45)(17,26,46)(18,27,47), (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( (1,28,48)(2,29,49)(3,30,50)(4,31,51)(5,32,52)(6,33,53)(7,34,54)(8,35,37)(9,36,38)(10,19,39)(11,20,40)(12,21,41)(13,22,42)(14,23,43)(15,24,44)(16,25,45)(17,26,46)(18,27,47), (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([[(1,28,48),(2,29,49),(3,30,50),(4,31,51),(5,32,52),(6,33,53),(7,34,54),(8,35,37),(9,36,38),(10,19,39),(11,20,40),(12,21,41),(13,22,42),(14,23,43),(15,24,44),(16,25,45),(17,26,46),(18,27,47)], [(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)]])
C3×C18 is a maximal subgroup of
C9⋊Dic3
54 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 9A | ··· | 9R | 18A | ··· | 18R |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 |
| kernel | C3×C18 | C3×C9 | C18 | C3×C6 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 18 | 18 |
Matrix representation of C3×C18 ►in GL2(𝔽19) generated by
| 7 | 0 |
| 0 | 11 |
| 5 | 0 |
| 0 | 3 |
G:=sub<GL(2,GF(19))| [7,0,0,11],[5,0,0,3] >;
C3×C18 in GAP, Magma, Sage, TeX
C_3\times C_{18} % in TeX
G:=Group("C3xC18"); // GroupNames label
G:=SmallGroup(54,9);
// by ID
G=gap.SmallGroup(54,9);
# by ID
G:=PCGroup([4,-2,-3,-3,-3,77]);
// Polycyclic
G:=Group<a,b|a^3=b^18=1,a*b=b*a>;
// generators/relations
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