metabelian, supersoluble, monomial, A-group
Aliases: C5⋊D15, C15⋊1D5, C52⋊3S3, C3⋊(C5⋊D5), (C5×C15)⋊1C2, SmallGroup(150,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C15 — C5⋊D15 |
Generators and relations for C5⋊D15
G = < a,b,c | a5=b15=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C5⋊D15
►On 75 points
Generators in S75
(1 53 37 22 62)(2 54 38 23 63)(3 55 39 24 64)(4 56 40 25 65)(5 57 41 26 66)(6 58 42 27 67)(7 59 43 28 68)(8 60 44 29 69)(9 46 45 30 70)(10 47 31 16 71)(11 48 32 17 72)(12 49 33 18 73)(13 50 34 19 74)(14 51 35 20 75)(15 52 36 21 61)
(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)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 45)(29 44)(30 43)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)
G:=sub<Sym(75)| (1,53,37,22,62)(2,54,38,23,63)(3,55,39,24,64)(4,56,40,25,65)(5,57,41,26,66)(6,58,42,27,67)(7,59,43,28,68)(8,60,44,29,69)(9,46,45,30,70)(10,47,31,16,71)(11,48,32,17,72)(12,49,33,18,73)(13,50,34,19,74)(14,51,35,20,75)(15,52,36,21,61), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,45)(29,44)(30,43)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)>;
G:=Group( (1,53,37,22,62)(2,54,38,23,63)(3,55,39,24,64)(4,56,40,25,65)(5,57,41,26,66)(6,58,42,27,67)(7,59,43,28,68)(8,60,44,29,69)(9,46,45,30,70)(10,47,31,16,71)(11,48,32,17,72)(12,49,33,18,73)(13,50,34,19,74)(14,51,35,20,75)(15,52,36,21,61), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,45)(29,44)(30,43)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69) );
G=PermutationGroup([[(1,53,37,22,62),(2,54,38,23,63),(3,55,39,24,64),(4,56,40,25,65),(5,57,41,26,66),(6,58,42,27,67),(7,59,43,28,68),(8,60,44,29,69),(9,46,45,30,70),(10,47,31,16,71),(11,48,32,17,72),(12,49,33,18,73),(13,50,34,19,74),(14,51,35,20,75),(15,52,36,21,61)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,45),(29,44),(30,43),(46,68),(47,67),(48,66),(49,65),(50,64),(51,63),(52,62),(53,61),(54,75),(55,74),(56,73),(57,72),(58,71),(59,70),(60,69)]])
C5⋊D15 is a maximal subgroup of
S3×C5⋊D5 D5×D15 C5⋊D45 C5⋊D15⋊C3 C15⋊D15
C5⋊D15 is a maximal quotient of C30.D5 C5⋊D45 C15⋊D15
39 conjugacy classes
| class | 1 | 2 | 3 | 5A | ··· | 5L | 15A | ··· | 15X |
| order | 1 | 2 | 3 | 5 | ··· | 5 | 15 | ··· | 15 |
| size | 1 | 75 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
39 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D5 | D15 |
| kernel | C5⋊D15 | C5×C15 | C52 | C15 | C5 |
| # reps | 1 | 1 | 1 | 12 | 24 |
Matrix representation of C5⋊D15 ►in GL4(𝔽31) generated by
| 0 | 1 | 0 | 0 |
| 30 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 30 | 12 |
| 5 | 16 | 0 | 0 |
| 15 | 11 | 0 | 0 |
| 0 | 0 | 16 | 5 |
| 0 | 0 | 26 | 14 |
| 5 | 16 | 0 | 0 |
| 14 | 26 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 19 | 1 |
G:=sub<GL(4,GF(31))| [0,30,0,0,1,12,0,0,0,0,0,30,0,0,1,12],[5,15,0,0,16,11,0,0,0,0,16,26,0,0,5,14],[5,14,0,0,16,26,0,0,0,0,30,19,0,0,0,1] >;
C5⋊D15 in GAP, Magma, Sage, TeX
C_5\rtimes D_{15} % in TeX
G:=Group("C5:D15"); // GroupNames label
G:=SmallGroup(150,12);
// by ID
G=gap.SmallGroup(150,12);
# by ID
G:=PCGroup([4,-2,-3,-5,-5,33,290,1923]);
// Polycyclic
G:=Group<a,b,c|a^5=b^15=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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