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

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

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

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

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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