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G = C5⋊D15order 150 = 2·3·52

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

metabelian, supersoluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C5×C15 — C5⋊D15
C1C5C52C5×C15 — C5⋊D15
C5×C15 — C5⋊D15
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
3C5⋊D5

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···5L15A···15X
order1235···515···15
size17522···22···2

39 irreducible representations

dim11222
type+++++
imageC1C2S3D5D15
kernelC5⋊D15C5×C15C52C15C5
# reps1111224

Matrix representation of C5⋊D15 in GL4(𝔽31) generated by

0100
301200
0001
003012
,
51600
151100
00165
002614
,
51600
142600
00300
00191
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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Subgroup lattice of C5⋊D15 in TeX

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