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## G = S3×C30order 180 = 22·32·5

### Direct product of C30 and S3

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

 Derived series C1 — C3 — S3×C30
 Chief series C1 — C3 — C15 — C3×C15 — S3×C15 — S3×C30
 Lower central C3 — S3×C30
 Upper central C1 — C30

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

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

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

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 6H 6I 10A 10B 10C 10D 10E ··· 10L 15A ··· 15H 15I ··· 15T 30A ··· 30H 30I ··· 30T 30U ··· 30AJ order 1 2 2 2 3 3 3 3 3 5 5 5 5 6 6 6 6 6 6 6 6 6 10 10 10 10 10 ··· 10 15 ··· 15 15 ··· 15 30 ··· 30 30 ··· 30 30 ··· 30 size 1 1 3 3 1 1 2 2 2 1 1 1 1 1 1 2 2 2 3 3 3 3 1 1 1 1 3 ··· 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 S3 D6 C3×S3 C5×S3 S3×C6 S3×C10 S3×C15 S3×C30 kernel S3×C30 S3×C15 C3×C30 S3×C10 S3×C6 C5×S3 C30 C3×S3 C3×C6 D6 S3 C6 C30 C15 C10 C6 C5 C3 C2 C1 # reps 1 2 1 2 4 4 2 8 4 8 16 8 1 1 2 4 2 4 8 8

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

 21 0 0 21
,
 25 0 0 5
,
 0 1 1 0
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

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