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G = C5xDic15  order 300 = 22·3·52

Direct product of C5 and Dic15

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

Aliases: C5xDic15, C15:3C20, C30.7D5, C30.1C10, C15:5Dic5, C10.4D15, C52:7Dic3, C6.(C5xD5), C3:(C5xDic5), C10.(C5xS3), C2.(C5xD15), (C5xC15):10C4, (C5xC10).2S3, (C5xC30).2C2, C5:2(C5xDic3), SmallGroup(300,19)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C15 — C5xDic15
Chief series
C1 — C5 — C15 — C30 — C5xC30 — C5xDic15
Lower central
C15 — C5xDic15
Upper central
C1 — C10

Generators and relations for C5xDic15
 G = < a,b,c | a5=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 80 in 32 conjugacy classes, 18 normal (all characteristic)
Quotients: C1, C2, C4, C5, S3, D5, C10, Dic3, Dic5, C20, C5xS3, D15, C5xD5, C5xDic3, Dic15, C5xDic5, C5xD15, C5xDic15
C1
C2
C3
C5
C5
2C5
2C5
15C4
C6
C10
C10
2C10
2C10
C15
C15
2C15
2C15
C52
5Dic3
3Dic5
15C20
C30
C30
2C30
2C30
C5xC10
C5xC15
Dic15
5C5xDic3
3C5xDic5
C5xC30
C5xDic15

Smallest permutation representation of C5xDic15
►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++++--+-
imageC1C2C4C5C10C20S3D5Dic3Dic5C5xS3D15C5xD5C5xDic3Dic15C5xDic5C5xD15C5xDic15
kernelC5xDic15C5xC30C5xC15Dic15C30C15C5xC10C30C52C15C10C10C6C5C5C3C2C1
# reps11244812124484481616

Matrix representation of C5xDic15 ►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] >;
 

C5xDic15 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 C5xDic15 in TeX

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