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G = C5×Dic15order 300 = 22·3·52

Direct product of C5 and Dic15

direct product, metacyclic, supersoluble, monomial, A-group

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

C1C15 — C5×Dic15
C1C5C15C30C5×C30 — C5×Dic15
C15 — C5×Dic15
C1C10

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

2C5
2C5
15C4
2C10
2C10
2C15
2C15
5Dic3
3Dic5
15C20
2C30
2C30
5C5×Dic3
3C5×Dic5

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 4A4B5A5B5C5D5E···5N 6 10A10B10C10D10E···10N15A···15X20A···20H30A···30X
order1234455555···561010101010···1015···1520···2030···30
size112151511112···2211112···22···215···152···2

90 irreducible representations

dim111111222222222222
type++++--+-
imageC1C2C4C5C10C20S3D5Dic3Dic5C5×S3D15C5×D5C5×Dic3Dic15C5×Dic5C5×D15C5×Dic15
kernelC5×Dic15C5×C30C5×C15Dic15C30C15C5×C10C30C52C15C10C10C6C5C5C3C2C1
# reps11244812124484481616

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

20
02
,
170
011
,
030
10
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

Export

Subgroup lattice of C5×Dic15 in TeX

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