direct product, abelian, monomial, 5-elementary
Aliases: C5×C15, SmallGroup(75,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C15 |
| C1 — C5×C15 |
| C1 — C5×C15 |
Generators and relations for C5×C15
G = < a,b | a5=b15=1, ab=ba >
Smallest permutation representation of C5×C15
►Regular action on 75 points
Generators in S75
(1 21 46 68 38)(2 22 47 69 39)(3 23 48 70 40)(4 24 49 71 41)(5 25 50 72 42)(6 26 51 73 43)(7 27 52 74 44)(8 28 53 75 45)(9 29 54 61 31)(10 30 55 62 32)(11 16 56 63 33)(12 17 57 64 34)(13 18 58 65 35)(14 19 59 66 36)(15 20 60 67 37)
(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)
G:=sub<Sym(75)| (1,21,46,68,38)(2,22,47,69,39)(3,23,48,70,40)(4,24,49,71,41)(5,25,50,72,42)(6,26,51,73,43)(7,27,52,74,44)(8,28,53,75,45)(9,29,54,61,31)(10,30,55,62,32)(11,16,56,63,33)(12,17,57,64,34)(13,18,58,65,35)(14,19,59,66,36)(15,20,60,67,37), (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)>;
G:=Group( (1,21,46,68,38)(2,22,47,69,39)(3,23,48,70,40)(4,24,49,71,41)(5,25,50,72,42)(6,26,51,73,43)(7,27,52,74,44)(8,28,53,75,45)(9,29,54,61,31)(10,30,55,62,32)(11,16,56,63,33)(12,17,57,64,34)(13,18,58,65,35)(14,19,59,66,36)(15,20,60,67,37), (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) );
G=PermutationGroup([[(1,21,46,68,38),(2,22,47,69,39),(3,23,48,70,40),(4,24,49,71,41),(5,25,50,72,42),(6,26,51,73,43),(7,27,52,74,44),(8,28,53,75,45),(9,29,54,61,31),(10,30,55,62,32),(11,16,56,63,33),(12,17,57,64,34),(13,18,58,65,35),(14,19,59,66,36),(15,20,60,67,37)], [(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)]])
C5×C15 is a maximal subgroup of
C5⋊D15 C52⋊C9
75 conjugacy classes
| class | 1 | 3A | 3B | 5A | ··· | 5X | 15A | ··· | 15AV |
| order | 1 | 3 | 3 | 5 | ··· | 5 | 15 | ··· | 15 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C5 | C15 |
| kernel | C5×C15 | C52 | C15 | C5 |
| # reps | 1 | 2 | 24 | 48 |
Matrix representation of C5×C15 ►in GL2(𝔽31) generated by
| 16 | 0 |
| 0 | 4 |
| 2 | 0 |
| 0 | 20 |
G:=sub<GL(2,GF(31))| [16,0,0,4],[2,0,0,20] >;
C5×C15 in GAP, Magma, Sage, TeX
C_5\times C_{15} % in TeX
G:=Group("C5xC15"); // GroupNames label
G:=SmallGroup(75,3);
// by ID
G=gap.SmallGroup(75,3);
# by ID
G:=PCGroup([3,-3,-5,-5]);
// Polycyclic
G:=Group<a,b|a^5=b^15=1,a*b=b*a>;
// generators/relations
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