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G = C15×Dic5order 300 = 22·3·52

Direct product of C15 and Dic5

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

Aliases: C15×Dic5, C52C60, C10.C30, C154C20, C528C12, C30.8D5, C30.2C10, C2.(D5×C15), (C5×C15)⋊11C4, C6.2(C5×D5), (C5×C10).3C6, (C5×C30).3C2, C10.4(C3×D5), SmallGroup(300,16)

Series: Derived Chief Lower central Upper central

C1C5 — C15×Dic5
C1C5C10C5×C10C5×C30 — C15×Dic5
C5 — C15×Dic5
C1C30

Generators and relations for C15×Dic5
 G = < a,b,c | a15=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

2C5
2C5
5C4
2C10
2C10
2C15
2C15
5C12
5C20
2C30
2C30
5C60

Smallest permutation representation of C15×Dic5
On 60 points
Generators in S60
(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 45 10 39 4 33 13 42 7 36)(2 31 11 40 5 34 14 43 8 37)(3 32 12 41 6 35 15 44 9 38)(16 46 22 52 28 58 19 49 25 55)(17 47 23 53 29 59 20 50 26 56)(18 48 24 54 30 60 21 51 27 57)
(1 50 33 23)(2 51 34 24)(3 52 35 25)(4 53 36 26)(5 54 37 27)(6 55 38 28)(7 56 39 29)(8 57 40 30)(9 58 41 16)(10 59 42 17)(11 60 43 18)(12 46 44 19)(13 47 45 20)(14 48 31 21)(15 49 32 22)

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

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

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

120 conjugacy classes

class 1  2 3A3B4A4B5A5B5C5D5E···5N6A6B10A10B10C10D10E···10N12A12B12C12D15A···15H15I···15AB20A···20H30A···30H30I···30AB60A···60P
order12334455555···5661010101010···101212121215···1515···1520···2030···3030···3060···60
size11115511112···21111112···255551···12···25···51···12···25···5

120 irreducible representations

dim11111111111122222222
type+++-
imageC1C2C3C4C5C6C10C12C15C20C30C60D5Dic5C3×D5C5×D5C3×Dic5C5×Dic5D5×C15C15×Dic5
kernelC15×Dic5C5×C30C5×Dic5C5×C15C3×Dic5C5×C10C30C52Dic5C15C10C5C30C15C10C6C5C3C2C1
# reps11224244888162248481616

Matrix representation of C15×Dic5 in GL2(𝔽31) generated by

190
019
,
230
027
,
026
250
G:=sub<GL(2,GF(31))| [19,0,0,19],[23,0,0,27],[0,25,26,0] >;

C15×Dic5 in GAP, Magma, Sage, TeX

C_{15}\times {\rm Dic}_5
% in TeX

G:=Group("C15xDic5");
// GroupNames label

G:=SmallGroup(300,16);
// by ID

G=gap.SmallGroup(300,16);
# by ID

G:=PCGroup([5,-2,-3,-5,-2,-5,150,6004]);
// Polycyclic

G:=Group<a,b,c|a^15=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C15×Dic5 in TeX

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