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G = C10xD15order 300 = 22·3·52

Direct product of C10 and D15

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

Aliases: C10xD15, C30:3D5, C30:1C10, C52:7D6, C15:8D10, C6:(C5xD5), C10:(C5xS3), C3:2(D5xC10), C5:2(S3xC10), (C5xC10):2S3, (C5xC30):2C2, C15:2(C2xC10), (C5xC15):7C22, SmallGroup(300,47)

Series: Derived Chief Lower central Upper central

C1C15 — C10xD15
C1C5C15C5xC15C5xD15 — C10xD15
C15 — C10xD15
C1C10

Generators and relations for C10xD15
 G = < a,b,c | a10=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 176 in 48 conjugacy classes, 22 normal (18 characteristic)
Quotients: C1, C2, C22, C5, S3, D5, C10, D6, D10, C2xC10, C5xS3, D15, C5xD5, S3xC10, D30, D5xC10, C5xD15, C10xD15
15C2
15C2
2C5
2C5
15C22
5S3
5S3
2C10
2C10
3D5
3D5
15C10
15C10
2C15
2C15
5D6
3D10
15C2xC10
2C30
2C30
5C5xS3
5C5xS3
3C5xD5
3C5xD5
5S3xC10
3D5xC10

Smallest permutation representation of C10xD15
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 46 34 49 37 52 40 55 43 58)(32 47 35 50 38 53 41 56 44 59)(33 48 36 51 39 54 42 57 45 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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 60)(9 59)(10 58)(11 57)(12 56)(13 55)(14 54)(15 53)(16 35)(17 34)(18 33)(19 32)(20 31)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)

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

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

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

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+++++++++
imageC1C2C2C5C10C10S3D5D6D10C5xS3D15C5xD5S3xC10D30D5xC10C5xD15C10xD15
kernelC10xD15C5xD15C5xC30D30D15C30C5xC10C30C52C15C10C10C6C5C5C3C2C1
# reps12148412124484481616

Matrix representation of C10xD15 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] >;

C10xD15 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 C10xD15 in TeX

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