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G = C6×D15order 180 = 22·32·5

Direct product of C6 and D15

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

Aliases: C6×D15, C301C6, C302S3, C157D6, C325D10, C6⋊(C3×D5), C10⋊(C3×S3), C52(S3×C6), (C3×C6)⋊1D5, C32(C6×D5), C152(C2×C6), (C3×C30)⋊2C2, (C3×C15)⋊7C22, SmallGroup(180,34)

Series: Derived Chief Lower central Upper central

C1C15 — C6×D15
C1C5C15C3×C15C3×D15 — C6×D15
C15 — C6×D15
C1C6

Generators and relations for C6×D15
 G = < a,b,c | a6=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

15C2
15C2
2C3
15C22
2C6
5S3
5S3
15C6
15C6
3D5
3D5
2C15
5D6
15C2×C6
5C3×S3
5C3×S3
3D10
2C30
3C3×D5
3C3×D5
5S3×C6
3C6×D5

Smallest permutation representation of C6×D15
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)]])

C6×D15 is a maximal subgroup of   D6⋊D15  C3⋊D60  D30.S3  D30⋊S3  C323D20  S3×C6×D5

54 conjugacy classes

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

54 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6S3D5D6C3×S3D10C3×D5D15S3×C6C6×D5D30C3×D15C6×D15
kernelC6×D15C3×D15C3×C30D30D15C30C30C3×C6C15C10C32C6C6C5C3C3C2C1
# reps121242121224424488

Matrix representation of C6×D15 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] >;

C6×D15 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 C6×D15 in TeX

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