Copied to
clipboard

G = C3×Dic15order 180 = 22·32·5

Direct product of C3 and Dic15

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

Aliases: C3×Dic15, C153C12, C30.1C6, C30.4S3, C6.4D15, C154Dic3, C322Dic5, C6.(C3×D5), (C3×C15)⋊8C4, C10.(C3×S3), C3⋊(C3×Dic5), C2.(C3×D15), (C3×C6).1D5, C52(C3×Dic3), (C3×C30).2C2, SmallGroup(180,15)

Series: Derived Chief Lower central Upper central

C1C15 — C3×Dic15
C1C5C15C30C3×C30 — C3×Dic15
C15 — C3×Dic15
C1C6

Generators and relations for C3×Dic15
 G = < a,b,c | a3=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >

2C3
15C4
2C6
2C15
5Dic3
15C12
3Dic5
2C30
5C3×Dic3
3C3×Dic5

Smallest permutation representation of C3×Dic15
On 60 points
Generators in S60
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 41 51)(32 42 52)(33 43 53)(34 44 54)(35 45 55)(36 46 56)(37 47 57)(38 48 58)(39 49 59)(40 50 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 41 16 56)(2 40 17 55)(3 39 18 54)(4 38 19 53)(5 37 20 52)(6 36 21 51)(7 35 22 50)(8 34 23 49)(9 33 24 48)(10 32 25 47)(11 31 26 46)(12 60 27 45)(13 59 28 44)(14 58 29 43)(15 57 30 42)

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

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

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

C3×Dic15 is a maximal subgroup of
C3×D5×Dic3  C3×S3×Dic5  C6.D30  D62D15  C3⋊Dic30  D30.S3  Dic15⋊S3  D30⋊S3  C323Dic10  C12×D15

54 conjugacy classes

class 1  2 3A3B3C3D3E4A4B5A5B6A6B6C6D6E10A10B12A12B12C12D15A···15P30A···30P
order123333344556666610101212121215···1530···30
size11112221515221122222151515152···22···2

54 irreducible representations

dim111111222222222222
type++++--+-
imageC1C2C3C4C6C12S3D5Dic3C3×S3Dic5C3×D5D15C3×Dic3C3×Dic5Dic15C3×D15C3×Dic15
kernelC3×Dic15C3×C30Dic15C3×C15C30C15C30C3×C6C15C10C32C6C6C5C3C3C2C1
# reps112224121224424488

Matrix representation of C3×Dic15 in GL2(𝔽31) generated by

50
05
,
130
012
,
030
10
G:=sub<GL(2,GF(31))| [5,0,0,5],[13,0,0,12],[0,1,30,0] >;

C3×Dic15 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{15}
% in TeX

G:=Group("C3xDic15");
// GroupNames label

G:=SmallGroup(180,15);
// by ID

G=gap.SmallGroup(180,15);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-5,30,483,3604]);
// Polycyclic

G:=Group<a,b,c|a^3=b^30=1,c^2=b^15,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic15 in TeX

׿
×
𝔽