direct product, abelian, monomial, 2-elementary
Aliases: C2×C30, SmallGroup(60,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C30 |
| C1 — C2×C30 |
| C1 — C2×C30 |
Generators and relations for C2×C30
G = < a,b | a2=b30=1, ab=ba >
Smallest permutation representation of C2×C30
►Regular action on 60 points
Generators in S60
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 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)
G:=sub<Sym(60)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50), (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)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50), (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) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50)], [(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)]])
C2×C30 is a maximal subgroup of
C15⋊7D4
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 10A | ··· | 10L | 15A | ··· | 15H | 30A | ··· | 30X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 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 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 |
| kernel | C2×C30 | C30 | C2×C10 | C2×C6 | C10 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 4 | 6 | 12 | 8 | 24 |
Matrix representation of C2×C30 ►in GL2(𝔽31) generated by
| 30 | 0 |
| 0 | 1 |
| 5 | 0 |
| 0 | 29 |
G:=sub<GL(2,GF(31))| [30,0,0,1],[5,0,0,29] >;
C2×C30 in GAP, Magma, Sage, TeX
C_2\times C_{30} % in TeX
G:=Group("C2xC30"); // GroupNames label
G:=SmallGroup(60,13);
// by ID
G=gap.SmallGroup(60,13);
# by ID
G:=PCGroup([4,-2,-2,-3,-5]);
// Polycyclic
G:=Group<a,b|a^2=b^30=1,a*b=b*a>;
// generators/relations
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