direct product, abelian, monomial, 3-elementary
Aliases: C3×C30, SmallGroup(90,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C30 |
| C1 — C3×C30 |
| C1 — C3×C30 |
Generators and relations for C3×C30
G = < a,b | a3=b30=1, ab=ba >
Smallest permutation representation of C3×C30
►Regular action on 90 points
Generators in S90
(1 47 76)(2 48 77)(3 49 78)(4 50 79)(5 51 80)(6 52 81)(7 53 82)(8 54 83)(9 55 84)(10 56 85)(11 57 86)(12 58 87)(13 59 88)(14 60 89)(15 31 90)(16 32 61)(17 33 62)(18 34 63)(19 35 64)(20 36 65)(21 37 66)(22 38 67)(23 39 68)(24 40 69)(25 41 70)(26 42 71)(27 43 72)(28 44 73)(29 45 74)(30 46 75)
(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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)
G:=sub<Sym(90)| (1,47,76)(2,48,77)(3,49,78)(4,50,79)(5,51,80)(6,52,81)(7,53,82)(8,54,83)(9,55,84)(10,56,85)(11,57,86)(12,58,87)(13,59,88)(14,60,89)(15,31,90)(16,32,61)(17,33,62)(18,34,63)(19,35,64)(20,36,65)(21,37,66)(22,38,67)(23,39,68)(24,40,69)(25,41,70)(26,42,71)(27,43,72)(28,44,73)(29,45,74)(30,46,75), (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)>;
G:=Group( (1,47,76)(2,48,77)(3,49,78)(4,50,79)(5,51,80)(6,52,81)(7,53,82)(8,54,83)(9,55,84)(10,56,85)(11,57,86)(12,58,87)(13,59,88)(14,60,89)(15,31,90)(16,32,61)(17,33,62)(18,34,63)(19,35,64)(20,36,65)(21,37,66)(22,38,67)(23,39,68)(24,40,69)(25,41,70)(26,42,71)(27,43,72)(28,44,73)(29,45,74)(30,46,75), (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90) );
G=PermutationGroup([[(1,47,76),(2,48,77),(3,49,78),(4,50,79),(5,51,80),(6,52,81),(7,53,82),(8,54,83),(9,55,84),(10,56,85),(11,57,86),(12,58,87),(13,59,88),(14,60,89),(15,31,90),(16,32,61),(17,33,62),(18,34,63),(19,35,64),(20,36,65),(21,37,66),(22,38,67),(23,39,68),(24,40,69),(25,41,70),(26,42,71),(27,43,72),(28,44,73),(29,45,74),(30,46,75)], [(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),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)]])
C3×C30 is a maximal subgroup of
C3⋊Dic15
90 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 6A | ··· | 6H | 10A | 10B | 10C | 10D | 15A | ··· | 15AF | 30A | ··· | 30AF |
| order | 1 | 2 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 |
| kernel | C3×C30 | C3×C15 | C30 | C3×C6 | C15 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 4 | 8 | 4 | 32 | 32 |
Matrix representation of C3×C30 ►in GL2(𝔽31) generated by
| 5 | 0 |
| 0 | 5 |
| 29 | 0 |
| 0 | 21 |
G:=sub<GL(2,GF(31))| [5,0,0,5],[29,0,0,21] >;
C3×C30 in GAP, Magma, Sage, TeX
C_3\times C_{30} % in TeX
G:=Group("C3xC30"); // GroupNames label
G:=SmallGroup(90,10);
// by ID
G=gap.SmallGroup(90,10);
# by ID
G:=PCGroup([4,-2,-3,-3,-5]);
// Polycyclic
G:=Group<a,b|a^3=b^30=1,a*b=b*a>;
// generators/relations
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