Copied to
clipboard

## G = S3×D20order 240 = 24·3·5

### Direct product of S3 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — S3×D20
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — S3×D20
 Lower central C15 — C30 — S3×D20
 Upper central C1 — C2 — C4

Generators and relations for S3×D20
G = < a,b,c,d | a3=b2=c20=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 616 in 108 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×9], C5, S3 [×2], S3 [×2], C6, C6 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×4], C10, C10 [×2], Dic3, C12, D6, D6 [×6], C2×C6 [×2], C15, C2×D4, C20, C20, D10 [×2], D10 [×6], C2×C10, C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, D20, D20 [×3], C2×C20, C22×D5 [×2], S3×D4, C5×Dic3, C60, S3×D5 [×4], C6×D5 [×2], S3×C10, D30 [×2], C2×D20, C3⋊D20 [×2], C3×D20, S3×C20, D60, C2×S3×D5 [×2], S3×D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, D20 [×2], C22×D5, S3×D4, S3×D5, C2×D20, C2×S3×D5, S3×D20

Smallest permutation representation of S3×D20
On 60 points
Generators in S60
(1 30 54)(2 31 55)(3 32 56)(4 33 57)(5 34 58)(6 35 59)(7 36 60)(8 37 41)(9 38 42)(10 39 43)(11 40 44)(12 21 45)(13 22 46)(14 23 47)(15 24 48)(16 25 49)(17 26 50)(18 27 51)(19 28 52)(20 29 53)
(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 41)(38 42)(39 43)(40 44)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 12 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 30A 30B 60A 60B 60C 60D order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 10 10 10 10 10 10 12 15 15 20 20 20 20 20 20 20 20 30 30 60 60 60 60 size 1 1 3 3 10 10 30 30 2 2 6 2 2 2 20 20 2 2 6 6 6 6 4 4 4 2 2 2 2 6 6 6 6 4 4 4 4 4 4

39 irreducible representations

 dim 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 S3 D4 D5 D6 D6 D10 D10 D10 D20 S3×D4 S3×D5 C2×S3×D5 S3×D20 kernel S3×D20 C3⋊D20 C3×D20 S3×C20 D60 C2×S3×D5 D20 C5×S3 C4×S3 C20 D10 Dic3 C12 D6 S3 C5 C4 C2 C1 # reps 1 2 1 1 1 2 1 2 2 1 2 2 2 2 8 1 2 2 4

Matrix representation of S3×D20 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 60 60 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 60 60
,
 36 32 0 0 2 34 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 19 60 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,1,0,0,60,0],[1,0,0,0,0,1,0,0,0,0,1,60,0,0,0,60],[36,2,0,0,32,34,0,0,0,0,1,0,0,0,0,1],[1,19,0,0,0,60,0,0,0,0,1,0,0,0,0,1] >;

S3×D20 in GAP, Magma, Sage, TeX

S_3\times D_{20}
% in TeX

G:=Group("S3xD20");
// GroupNames label

G:=SmallGroup(240,137);
// by ID

G=gap.SmallGroup(240,137);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,50,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^20=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽