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## G = S3×C5⋊D4order 240 = 24·3·5

### Direct product of S3 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — S3×C5⋊D4
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — S3×C5⋊D4
 Lower central C15 — C30 — S3×C5⋊D4
 Upper central C1 — C2 — C22

Generators and relations for S3×C5⋊D4
G = < a,b,c,d,e | a3=b2=c5=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 504 in 108 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2 [×6], C3, C4 [×2], C22, C22 [×8], C5, S3 [×2], S3 [×2], C6, C6 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×2], C10, C10 [×4], Dic3, C12, D6 [×2], D6 [×5], C2×C6, C2×C6, C15, C2×D4, Dic5, Dic5, D10, D10 [×3], C2×C10, C2×C10 [×4], C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3, C22×S3, C5×S3 [×2], C5×S3, C3×D5, D15, C30, C30, C2×Dic5, C5⋊D4, C5⋊D4 [×3], C22×D5, C22×C10, S3×D4, C3×Dic5, Dic15, S3×D5 [×2], C6×D5, S3×C10 [×2], S3×C10 [×2], D30, C2×C30, C2×C5⋊D4, S3×Dic5, C15⋊D4, C5⋊D12, C3×C5⋊D4, C157D4, C2×S3×D5, S3×C2×C10, S3×C5⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, C2×S3×D5, S3×C5⋊D4

Smallest permutation representation of S3×C5⋊D4
On 60 points
Generators in S60
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(36 41)(37 42)(38 43)(39 44)(40 45)(51 56)(52 57)(53 58)(54 59)(55 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 31 16 46)(2 35 17 50)(3 34 18 49)(4 33 19 48)(5 32 20 47)(6 36 21 51)(7 40 22 55)(8 39 23 54)(9 38 24 53)(10 37 25 52)(11 41 26 56)(12 45 27 60)(13 44 28 59)(14 43 29 58)(15 42 30 57)
(2 5)(3 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(31 46)(32 50)(33 49)(34 48)(35 47)(36 51)(37 55)(38 54)(39 53)(40 52)(41 56)(42 60)(43 59)(44 58)(45 57)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 10A ··· 10F 10G ··· 10N 12 15A 15B 30A ··· 30F order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 10 ··· 10 10 ··· 10 12 15 15 30 ··· 30 size 1 1 2 3 3 6 10 30 2 10 30 2 2 2 4 20 2 ··· 2 6 ··· 6 20 4 4 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D6 D10 D10 C5⋊D4 S3×D4 S3×D5 C2×S3×D5 S3×C5⋊D4 kernel S3×C5⋊D4 S3×Dic5 C15⋊D4 C5⋊D12 C3×C5⋊D4 C15⋊7D4 C2×S3×D5 S3×C2×C10 C5⋊D4 C5×S3 C22×S3 Dic5 D10 C2×C10 D6 C2×C6 S3 C5 C22 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 4 2 8 1 2 2 4

Matrix representation of S3×C5⋊D4 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 0 60 0 0 1 60
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 0 60 0 0 1 17 0 0 0 0 1 0 0 0 0 1
,
 22 22 0 0 14 39 0 0 0 0 1 0 0 0 0 1
,
 17 44 0 0 60 44 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,60],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,60,17,0,0,0,0,1,0,0,0,0,1],[22,14,0,0,22,39,0,0,0,0,1,0,0,0,0,1],[17,60,0,0,44,44,0,0,0,0,1,0,0,0,0,1] >;

S3×C5⋊D4 in GAP, Magma, Sage, TeX

S_3\times C_5\rtimes D_4
% in TeX

G:=Group("S3xC5:D4");
// GroupNames label

G:=SmallGroup(240,150);
// by ID

G=gap.SmallGroup(240,150);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,490,6917]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^5=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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