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G = C10×D15order 300 = 22·3·52

Direct product of C10 and D15

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

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

C1C15 — C10×D15
C1C5C15C5×C15C5×D15 — C10×D15
C15 — C10×D15
C1C10

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

15C2
15C2
2C5
2C5
15C22
5S3
5S3
2C10
2C10
3D5
3D5
15C10
15C10
2C15
2C15
5D6
3D10
15C2×C10
2C30
2C30
5C5×S3
5C5×S3
3C5×D5
3C5×D5
5S3×C10
3D5×C10

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

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

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

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

90 conjugacy classes

class 1 2A2B2C 3 5A5B5C5D5E···5N 6 10A10B10C10D10E···10N10O···10V15A···15X30A···30X
order1222355555···561010101010···1010···1015···1530···30
size111515211112···2211112···215···152···22···2

90 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C5C10C10S3D5D6D10C5×S3D15C5×D5S3×C10D30D5×C10C5×D15C10×D15
kernelC10×D15C5×D15C5×C30D30D15C30C5×C10C30C52C15C10C10C6C5C5C3C2C1
# reps12148412124484481616

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

290
029
,
814
630
,
300
251
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

Export

Subgroup lattice of C10×D15 in TeX

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