S3xD20 - GroupNames

G = S₃xD₂₀ order 240 = 2⁴·3·5

Direct product of S₃ and D₂₀

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

Aliases: S₃xD₂₀, C₂₀:₄D₆, D₁₀:₁D₆, D₆₀:₇C₂, C₁₂:₁D₁₀, C₆₀:₂C₂², Dic₃:₃D₁₀, D₆.₁₀D₁₀, D₃₀:₂C₂², C₃₀.₁₃C₂³, C₅:₁(S₃xD₄), C₄:₁(S₃xD₅), C₃:₁(C₂xD₂₀), C₁₅:₂(C₂xD₄), (C₄xS₃):₃D₅, (C₅xS₃):₁D₄, (S₃xC₂₀):₃C₂, (C₃xD₂₀):₃C₂, C₃:D₂₀:₃C₂, (C₆xD₅):₁C₂², C₆.₁₃(C₂²xD₅), C₁₀.₁₃(C₂²xS₃), (C₅xDic₃):₄C₂², (S₃xC₁₀).₁₀C₂², (C₂xS₃xD₅):₂C₂, C₂.₁₆(C₂xS₃xD₅), SmallGroup(240,137)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — S₃xD₂₀

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₆xD₅ — C₂xS₃xD₅ — S₃xD₂₀

Lower central

C₁₅ — C₃₀ — S₃xD₂₀

Upper central

C₁ — C₂ — C₄

Generators and relations for S₃xD₂₀
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, C₃, C₄, C₄, C₂², C₅, S₃, S₃, C₆, C₆, C₂xC₄, D₄, C₂³, D₅, C₁₀, C₁₀, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₁₅, C₂xD₄, C₂₀, C₂₀, D₁₀, D₁₀, C₂xC₁₀, C₄xS₃, D₁₂, C₃:D₄, C₃xD₄, C₂²xS₃, C₅xS₃, C₃xD₅, D₁₅, C₃₀, D₂₀, D₂₀, C₂xC₂₀, C₂²xD₅, S₃xD₄, C₅xDic₃, C₆₀, S₃xD₅, C₆xD₅, S₃xC₁₀, D₃₀, C₂xD₂₀, C₃:D₂₀, C₃xD₂₀, S₃xC₂₀, D₆₀, C₂xS₃xD₅, S₃xD₂₀
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₅, D₆, C₂xD₄, D₁₀, C₂²xS₃, D₂₀, C₂²xD₅, S₃xD₄, S₃xD₅, C₂xD₂₀, C₂xS₃xD₅, S₃xD₂₀

Smallest permutation representation of S₃xD₂₀
►On 60 points

Generators in S₆₀

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

(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 41)(40 42)

(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 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)

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

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

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

S₃xD₂₀ is a maximal subgroup of
C₄₀:₁D₆ D₄₀:S₃ D₆₀.C₂² D₁₂:D₁₀ D₂₀:₂₅D₆ D₂₀:₂₉D₆ S₃xD₄xD₅ D₂₀:₁₄D₆ D₂₀:₁₇D₆
S₃xD₂₀ is a maximal quotient of
C₄₀:₁D₆ D₄₀:S₃ Dic₂₀:S₃ D₆.₁D₂₀ D₄₀:₇S₃ C₄₀.₂D₆ D₁₂₀:₅C₂ Dic₃.D₂₀ Dic₃:₄D₂₀ Dic₃:D₂₀ D₁₀:₂Dic₆ D₆.D₂₀ D₆₀:₁₄C₄ D₃₀:₄Q₈ D₆:D₂₀ C₆₀:₄D₄ D₆.₉D₂₀ C₁₂:D₂₀ D₃₀:₂D₄ C₆₀:₆D₄ C₂₀:₄Dic₆ D₆:₄D₂₀ D₃₀:₅D₄

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

type

image

C₁

C₂

S₃

D₄

D₅

D₆

D₁₀

D₂₀

S₃xD₄

S₃xD₅

C₂xS₃xD₅

S₃xD₂₀

kernel

S₃xD₂₀

C₃:D₂₀

C₃xD₂₀

S₃xC₂₀

D₆₀

C₂xS₃xD₅

D₂₀

C₅xS₃

C₄xS₃

C₂₀

D₁₀

Dic₃

C₁₂

D₆

S₃

C₅

C₄

C₂

C₁

# reps

Matrix representation of S₃xD₂₀ ►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] >;

S₃xD₂₀ 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

G = S3xD20 order 240 = 24·3·5

Direct product of S3 and D20

G = S₃xD₂₀ order 240 = 2⁴·3·5

Direct product of S₃ and D₂₀