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## G = C10×D15order 300 = 22·3·52

### Direct product of C10 and D15

Aliases: C10×D15, C303D5, C301C10, C527D6, C158D10, C6⋊(C5×D5), C10⋊(C5×S3), C32(D5×C10), C52(S3×C10), (C5×C10)⋊2S3, (C5×C30)⋊2C2, C152(C2×C10), (C5×C15)⋊7C22, SmallGroup(300,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C10×D15
 Chief series C1 — C5 — C15 — C5×C15 — C5×D15 — C10×D15
 Lower central C15 — C10×D15
 Upper central C1 — C10

Generators and relations for C10×D15
G = < a,b,c | a10=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C10×D15
On 60 points
Generators in S60
(1 26 13 23 10 20 7 17 4 29)(2 27 14 24 11 21 8 18 5 30)(3 28 15 25 12 22 9 19 6 16)(31 46 34 49 37 52 40 55 43 58)(32 47 35 50 38 53 41 56 44 59)(33 48 36 51 39 54 42 57 45 60)
(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)
(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 60)(9 59)(10 58)(11 57)(12 56)(13 55)(14 54)(15 53)(16 35)(17 34)(18 33)(19 32)(20 31)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3 5A 5B 5C 5D 5E ··· 5N 6 10A 10B 10C 10D 10E ··· 10N 10O ··· 10V 15A ··· 15X 30A ··· 30X order 1 2 2 2 3 5 5 5 5 5 ··· 5 6 10 10 10 10 10 ··· 10 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 15 15 2 1 1 1 1 2 ··· 2 2 1 1 1 1 2 ··· 2 15 ··· 15 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C5 C10 C10 S3 D5 D6 D10 C5×S3 D15 C5×D5 S3×C10 D30 D5×C10 C5×D15 C10×D15 kernel C10×D15 C5×D15 C5×C30 D30 D15 C30 C5×C10 C30 C52 C15 C10 C10 C6 C5 C5 C3 C2 C1 # reps 1 2 1 4 8 4 1 2 1 2 4 4 8 4 4 8 16 16

Matrix representation of C10×D15 in GL2(𝔽31) generated by

 29 0 0 29
,
 8 14 6 30
,
 30 0 25 1
G:=sub<GL(2,GF(31))| [29,0,0,29],[8,6,14,30],[30,25,0,1] >;

C10×D15 in GAP, Magma, Sage, TeX

C_{10}\times D_{15}
% in TeX

G:=Group("C10xD15");
// GroupNames label

G:=SmallGroup(300,47);
// by ID

G=gap.SmallGroup(300,47);
# by ID

G:=PCGroup([5,-2,-2,-5,-3,-5,803,6004]);
// Polycyclic

G:=Group<a,b,c|a^10=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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