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## G = S3×C2×C10order 120 = 23·3·5

### Direct product of C2×C10 and S3

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

 Derived series C1 — C3 — S3×C2×C10
 Chief series C1 — C3 — C15 — C5×S3 — S3×C10 — S3×C2×C10
 Lower central C3 — S3×C2×C10
 Upper central C1 — C2×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 2A 2B 2C 2D 2E 2F 2G 3 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M ··· 10AB 15A 15B 15C 15D 30A ··· 30L order 1 2 2 2 2 2 2 2 3 5 5 5 5 6 6 6 10 ··· 10 10 ··· 10 15 15 15 15 30 ··· 30 size 1 1 1 1 3 3 3 3 2 1 1 1 1 2 2 2 1 ··· 1 3 ··· 3 2 2 2 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C5 C10 C10 S3 D6 C5×S3 S3×C10 kernel S3×C2×C10 S3×C10 C2×C30 C22×S3 D6 C2×C6 C2×C10 C10 C22 C2 # reps 1 6 1 4 24 4 1 3 4 12

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

 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 30 0 0 1 30
,
 1 0 0 0 0 1 0 0 0 0 1 30 0 0 0 30
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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