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G = S3×C2×C10order 120 = 23·3·5

Direct product of C2×C10 and S3

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×C2×C10, C154C23, C304C22, C6⋊(C2×C10), C3⋊(C22×C10), (C2×C30)⋊7C2, (C2×C6)⋊3C10, SmallGroup(120,45)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C2×C10
C1C3C15C5×S3S3×C10 — S3×C2×C10
C3 — S3×C2×C10
C1C2×C10

Generators and relations for S3×C2×C10
 G = < a,b,c,d | a2=b10=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 108 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2 [×3], C2 [×4], C3, C22, C22 [×6], C5, S3 [×4], C6 [×3], C23, C10 [×3], C10 [×4], D6 [×6], C2×C6, C15, C2×C10, C2×C10 [×6], C22×S3, C5×S3 [×4], C30 [×3], C22×C10, S3×C10 [×6], C2×C30, S3×C2×C10
Quotients: C1, C2 [×7], C22 [×7], C5, S3, C23, C10 [×7], D6 [×3], C2×C10 [×7], C22×S3, C5×S3, C22×C10, S3×C10 [×3], S3×C2×C10

Smallest permutation representation of S3×C2×C10
On 60 points
Generators in S60
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 48)(12 49)(13 50)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 40)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)
(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 42 39)(2 43 40)(3 44 31)(4 45 32)(5 46 33)(6 47 34)(7 48 35)(8 49 36)(9 50 37)(10 41 38)(11 26 57)(12 27 58)(13 28 59)(14 29 60)(15 30 51)(16 21 52)(17 22 53)(18 23 54)(19 24 55)(20 25 56)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 31)(18 32)(19 33)(20 34)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 41)(30 42)

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

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

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

S3×C2×C10 is a maximal subgroup of   D6⋊Dic5

60 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 5A5B5C5D6A6B6C10A···10L10M···10AB15A15B15C15D30A···30L
order122222223555566610···1010···101515151530···30
size11113333211112221···13···322222···2

60 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10S3D6C5×S3S3×C10
kernelS3×C2×C10S3×C10C2×C30C22×S3D6C2×C6C2×C10C10C22C2
# reps161424413412

Matrix representation of S3×C2×C10 in GL4(𝔽31) generated by

30000
03000
0010
0001
,
30000
0100
0020
0002
,
1000
0100
00030
00130
,
1000
0100
00130
00030
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[30,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,30,30],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,30,30] >;

S3×C2×C10 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{10}
% in TeX

G:=Group("S3xC2xC10");
// GroupNames label

G:=SmallGroup(120,45);
// by ID

G=gap.SmallGroup(120,45);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-3,2004]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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