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## G = C5×Dic15order 300 = 22·3·52

### Direct product of C5 and Dic15

Aliases: C5×Dic15, C153C20, C30.7D5, C30.1C10, C155Dic5, C10.4D15, C527Dic3, C6.(C5×D5), C3⋊(C5×Dic5), C10.(C5×S3), C2.(C5×D15), (C5×C15)⋊10C4, (C5×C10).2S3, (C5×C30).2C2, C52(C5×Dic3), SmallGroup(300,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C5×Dic15
 Chief series C1 — C5 — C15 — C30 — C5×C30 — C5×Dic15
 Lower central C15 — C5×Dic15
 Upper central C1 — C10

Generators and relations for C5×Dic15
G = < a,b,c | a5=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C5×Dic15
On 60 points
Generators in S60
(1 25 19 13 7)(2 26 20 14 8)(3 27 21 15 9)(4 28 22 16 10)(5 29 23 17 11)(6 30 24 18 12)(31 37 43 49 55)(32 38 44 50 56)(33 39 45 51 57)(34 40 46 52 58)(35 41 47 53 59)(36 42 48 54 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 36 16 51)(2 35 17 50)(3 34 18 49)(4 33 19 48)(5 32 20 47)(6 31 21 46)(7 60 22 45)(8 59 23 44)(9 58 24 43)(10 57 25 42)(11 56 26 41)(12 55 27 40)(13 54 28 39)(14 53 29 38)(15 52 30 37)

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

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

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

90 conjugacy classes

 class 1 2 3 4A 4B 5A 5B 5C 5D 5E ··· 5N 6 10A 10B 10C 10D 10E ··· 10N 15A ··· 15X 20A ··· 20H 30A ··· 30X order 1 2 3 4 4 5 5 5 5 5 ··· 5 6 10 10 10 10 10 ··· 10 15 ··· 15 20 ··· 20 30 ··· 30 size 1 1 2 15 15 1 1 1 1 2 ··· 2 2 1 1 1 1 2 ··· 2 2 ··· 2 15 ··· 15 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 C4 C5 C10 C20 S3 D5 Dic3 Dic5 C5×S3 D15 C5×D5 C5×Dic3 Dic15 C5×Dic5 C5×D15 C5×Dic15 kernel C5×Dic15 C5×C30 C5×C15 Dic15 C30 C15 C5×C10 C30 C52 C15 C10 C10 C6 C5 C5 C3 C2 C1 # reps 1 1 2 4 4 8 1 2 1 2 4 4 8 4 4 8 16 16

Matrix representation of C5×Dic15 in GL2(𝔽31) generated by

 2 0 0 2
,
 17 0 0 11
,
 0 30 1 0
G:=sub<GL(2,GF(31))| [2,0,0,2],[17,0,0,11],[0,1,30,0] >;

C5×Dic15 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_{15}
% in TeX

G:=Group("C5xDic15");
// GroupNames label

G:=SmallGroup(300,19);
// by ID

G=gap.SmallGroup(300,19);
# by ID

G:=PCGroup([5,-2,-5,-2,-3,-5,50,803,6004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^30=1,c^2=b^15,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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