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G = C6xD15order 180 = 22·32·5

Direct product of C6 and D15

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

Aliases: C6xD15, C30:1C6, C30:2S3, C15:7D6, C32:5D10, C6:(C3xD5), C10:(C3xS3), C5:2(S3xC6), (C3xC6):1D5, C3:2(C6xD5), C15:2(C2xC6), (C3xC30):2C2, (C3xC15):7C22, SmallGroup(180,34)

Series: Derived Chief Lower central Upper central

C1C15 — C6xD15
C1C5C15C3xC15C3xD15 — C6xD15
C15 — C6xD15
C1C6

Generators and relations for C6xD15
 G = < a,b,c | a6=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 168 in 44 conjugacy classes, 22 normal (18 characteristic)
Quotients: C1, C2, C3, C22, S3, C6, D5, D6, C2xC6, C3xS3, D10, C3xD5, D15, S3xC6, C6xD5, D30, C3xD15, C6xD15
15C2
15C2
2C3
15C22
2C6
5S3
5S3
15C6
15C6
3D5
3D5
2C15
5D6
15C2xC6
5C3xS3
5C3xS3
3D10
2C30
3C3xD5
3C3xD5
5S3xC6
3C6xD5

Smallest permutation representation of C6xD15
On 60 points
Generators in S60
(1 27 6 17 11 22)(2 28 7 18 12 23)(3 29 8 19 13 24)(4 30 9 20 14 25)(5 16 10 21 15 26)(31 49 41 59 36 54)(32 50 42 60 37 55)(33 51 43 46 38 56)(34 52 44 47 39 57)(35 53 45 48 40 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 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 45)(29 44)(30 43)

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

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

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

C6xD15 is a maximal subgroup of   D6:D15  C3:D60  D30.S3  D30:S3  C32:3D20  S3xC6xD5

54 conjugacy classes

class 1 2A2B2C3A3B3C3D3E5A5B6A6B6C6D6E6F6G6H6I10A10B15A···15P30A···30P
order12223333355666666666101015···1530···30
size11151511222221122215151515222···22···2

54 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6S3D5D6C3xS3D10C3xD5D15S3xC6C6xD5D30C3xD15C6xD15
kernelC6xD15C3xD15C3xC30D30D15C30C30C3xC6C15C10C32C6C6C5C3C3C2C1
# reps121242121224424488

Matrix representation of C6xD15 in GL2(F31) generated by

60
06
,
280
2210
,
48
227
G:=sub<GL(2,GF(31))| [6,0,0,6],[28,22,0,10],[4,2,8,27] >;

C6xD15 in GAP, Magma, Sage, TeX

C_6\times D_{15}
% in TeX

G:=Group("C6xD15");
// GroupNames label

G:=SmallGroup(180,34);
// by ID

G=gap.SmallGroup(180,34);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-5,483,3604]);
// Polycyclic

G:=Group<a,b,c|a^6=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6xD15 in TeX

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