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G = Dic3×C15order 180 = 22·32·5

Direct product of C15 and Dic3

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

Aliases: Dic3×C15, C3⋊C60, C6.C30, C155C12, C30.5C6, C30.8S3, C322C20, C2.(S3×C15), (C3×C15)⋊10C4, C6.4(C5×S3), C10.2(C3×S3), (C3×C6).1C10, (C3×C30).4C2, SmallGroup(180,14)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C15
C1C3C6C30C3×C30 — Dic3×C15
C3 — Dic3×C15
C1C30

Generators and relations for Dic3×C15
 G = < a,b,c | a15=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

2C3
3C4
2C6
2C15
3C12
3C20
2C30
3C60

Smallest permutation representation of Dic3×C15
On 60 points
Generators in S60
(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 11 46 6 56)(2 52 12 47 7 57)(3 53 13 48 8 58)(4 54 14 49 9 59)(5 55 15 50 10 60)(16 33 21 38 26 43)(17 34 22 39 27 44)(18 35 23 40 28 45)(19 36 24 41 29 31)(20 37 25 42 30 32)
(1 41 46 19)(2 42 47 20)(3 43 48 21)(4 44 49 22)(5 45 50 23)(6 31 51 24)(7 32 52 25)(8 33 53 26)(9 34 54 27)(10 35 55 28)(11 36 56 29)(12 37 57 30)(13 38 58 16)(14 39 59 17)(15 40 60 18)

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

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

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

Dic3×C15 is a maximal subgroup of   C6.D30  C3⋊D60  C3⋊Dic30  S3×C60

90 conjugacy classes

class 1  2 3A3B3C3D3E4A4B5A5B5C5D6A6B6C6D6E10A10B10C10D12A12B12C12D15A···15H15I···15T20A···20H30A···30H30I···30T60A···60P
order123333344555566666101010101212121215···1515···1520···2030···3030···3060···60
size111122233111111222111133331···12···23···31···12···23···3

90 irreducible representations

dim11111111111122222222
type+++-
imageC1C2C3C4C5C6C10C12C15C20C30C60S3Dic3C3×S3C5×S3C3×Dic3C5×Dic3S3×C15Dic3×C15
kernelDic3×C15C3×C30C5×Dic3C3×C15C3×Dic3C30C3×C6C15Dic3C32C6C3C30C15C10C6C5C3C2C1
# reps112242448881611242488

Matrix representation of Dic3×C15 in GL2(𝔽31) generated by

200
020
,
260
06
,
030
10
G:=sub<GL(2,GF(31))| [20,0,0,20],[26,0,0,6],[0,1,30,0] >;

Dic3×C15 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{15}
% in TeX

G:=Group("Dic3xC15");
// GroupNames label

G:=SmallGroup(180,14);
// by ID

G=gap.SmallGroup(180,14);
# by ID

G:=PCGroup([5,-2,-3,-5,-2,-3,150,3004]);
// Polycyclic

G:=Group<a,b,c|a^15=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Dic3×C15 in TeX

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