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## G = C5⋊D15order 150 = 2·3·52

### The semidirect product of C5 and D15 acting via D15/C15=C2

Aliases: C5⋊D15, C151D5, C523S3, C3⋊(C5⋊D5), (C5×C15)⋊1C2, SmallGroup(150,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C15 — C5⋊D15
 Chief series C1 — C5 — C52 — C5×C15 — C5⋊D15
 Lower central C5×C15 — C5⋊D15
 Upper central C1

Generators and relations for C5⋊D15
G = < a,b,c | a5=b15=c2=1, ab=ba, cac=a-1, cbc=b-1 >

75C2
25S3
15D5
15D5
15D5
15D5
15D5
15D5
5D15
5D15
5D15
5D15
5D15
5D15

Smallest permutation representation of C5⋊D15
On 75 points
Generators in S75
(1 60 34 29 69)(2 46 35 30 70)(3 47 36 16 71)(4 48 37 17 72)(5 49 38 18 73)(6 50 39 19 74)(7 51 40 20 75)(8 52 41 21 61)(9 53 42 22 62)(10 54 43 23 63)(11 55 44 24 64)(12 56 45 25 65)(13 57 31 26 66)(14 58 32 27 67)(15 59 33 28 68)
(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 31)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)

G:=sub<Sym(75)| (1,60,34,29,69)(2,46,35,30,70)(3,47,36,16,71)(4,48,37,17,72)(5,49,38,18,73)(6,50,39,19,74)(7,51,40,20,75)(8,52,41,21,61)(9,53,42,22,62)(10,54,43,23,63)(11,55,44,24,64)(12,56,45,25,65)(13,57,31,26,66)(14,58,32,27,67)(15,59,33,28,68), (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)(16,31)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)>;

G:=Group( (1,60,34,29,69)(2,46,35,30,70)(3,47,36,16,71)(4,48,37,17,72)(5,49,38,18,73)(6,50,39,19,74)(7,51,40,20,75)(8,52,41,21,61)(9,53,42,22,62)(10,54,43,23,63)(11,55,44,24,64)(12,56,45,25,65)(13,57,31,26,66)(14,58,32,27,67)(15,59,33,28,68), (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)(16,31)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68) );

G=PermutationGroup([(1,60,34,29,69),(2,46,35,30,70),(3,47,36,16,71),(4,48,37,17,72),(5,49,38,18,73),(6,50,39,19,74),(7,51,40,20,75),(8,52,41,21,61),(9,53,42,22,62),(10,54,43,23,63),(11,55,44,24,64),(12,56,45,25,65),(13,57,31,26,66),(14,58,32,27,67),(15,59,33,28,68)], [(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),(16,31),(17,45),(18,44),(19,43),(20,42),(21,41),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(46,67),(47,66),(48,65),(49,64),(50,63),(51,62),(52,61),(53,75),(54,74),(55,73),(56,72),(57,71),(58,70),(59,69),(60,68)])

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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