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G = S3×C30order 180 = 22·32·5

Direct product of C30 and S3

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

Aliases: S3×C30, C6⋊C30, C303C6, C3⋊(C2×C30), C154(C2×C6), (C3×C6)⋊1C10, (C3×C30)⋊4C2, C322(C2×C10), (C3×C15)⋊9C22, SmallGroup(180,33)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C30
C1C3C15C3×C15S3×C15 — S3×C30
C3 — S3×C30
C1C30

Generators and relations for S3×C30
 G = < a,b,c | a30=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
2C3
3C22
2C6
3C6
3C6
3C10
3C10
2C15
3C2×C6
3C2×C10
2C30
3C30
3C30
3C2×C30

Smallest permutation representation of S3×C30
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 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 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)

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

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

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

S3×C30 is a maximal subgroup of   D6⋊D15  D62D15

90 conjugacy classes

class 1 2A2B2C3A3B3C3D3E5A5B5C5D6A6B6C6D6E6F6G6H6I10A10B10C10D10E···10L15A···15H15I···15T30A···30H30I···30T30U···30AJ
order12223333355556666666661010101010···1015···1515···1530···3030···3030···30
size113311222111111222333311113···31···12···21···12···23···3

90 irreducible representations

dim11111111111122222222
type+++++
imageC1C2C2C3C5C6C6C10C10C15C30C30S3D6C3×S3C5×S3S3×C6S3×C10S3×C15S3×C30
kernelS3×C30S3×C15C3×C30S3×C10S3×C6C5×S3C30C3×S3C3×C6D6S3C6C30C15C10C6C5C3C2C1
# reps121244284816811242488

Matrix representation of S3×C30 in GL2(𝔽31) generated by

210
021
,
250
05
,
01
10
G:=sub<GL(2,GF(31))| [21,0,0,21],[25,0,0,5],[0,1,1,0] >;

S3×C30 in GAP, Magma, Sage, TeX

S_3\times C_{30}
% in TeX

G:=Group("S3xC30");
// GroupNames label

G:=SmallGroup(180,33);
// by ID

G=gap.SmallGroup(180,33);
# by ID

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

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

Export

Subgroup lattice of S3×C30 in TeX

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