S3^2xD5 - GroupNames

G = S₃²×D₅ order 360 = 2³·3²·5

Direct product of S₃, S₃ and D₅

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

Aliases: S₃²×D₅, D₁₅⋊₁D₆, (S₃×D₁₅)⋊C₂, (C₃×C₁₅)⋊C₂³, C₃⋊S₃⋊₁D₁₀, (C₃×D₅)⋊₁D₆, (C₅×S₃)⋊₁D₆, C₃⋊D₁₅⋊C₂², D₁₅⋊S₃⋊₃C₂, (C₃×S₃)⋊₁D₁₀, (S₃×C₁₅)⋊C₂², (C₃×D₁₅)⋊C₂², C₁₅⋊₁(C₂²×S₃), (C₃²×D₅)⋊C₂², C₃²⋊₁(C₂²×D₅), C₅⋊₁(C₂×S₃²), (C₃×S₃×D₅)⋊C₂, (D₅×C₃⋊S₃)⋊C₂, C₃⋊₁(C₂×S₃×D₅), (C₅×S₃²)⋊₁C₂, (C₅×C₃⋊S₃)⋊C₂², SmallGroup(360,137)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₅ — S₃²×D₅

Chief series

C₁ — C₅ — C₁₅ — C₃×C₁₅ — C₃²×D₅ — C₃×S₃×D₅ — S₃²×D₅

Lower central

C₃×C₁₅ — S₃²×D₅

Upper central

C₁

Generators and relations for S₃²×D₅
G = < a,b,c,d,e,f | a³=b²=c³=d²=e⁵=f²=1, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 996 in 138 conjugacy classes, 38 normal (14 characteristic)
C₁, C₂, C₃, C₃, C₂², C₅, S₃, S₃, C₆, C₂³, C₃², D₅, D₅, C₁₀, D₆, C₂×C₆, C₁₅, C₁₅, C₃×S₃, C₃×S₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, D₁₀, C₂×C₁₀, C₂²×S₃, C₅×S₃, C₅×S₃, C₃×D₅, C₃×D₅, D₁₅, D₁₅, C₃₀, S₃², S₃², S₃×C₆, C₂×C₃⋊S₃, C₂²×D₅, C₃×C₁₅, S₃×D₅, S₃×D₅, C₆×D₅, S₃×C₁₀, D₃₀, C₂×S₃², C₃²×D₅, S₃×C₁₅, C₃×D₁₅, C₅×C₃⋊S₃, C₃⋊D₁₅, C₂×S₃×D₅, C₃×S₃×D₅, D₅×C₃⋊S₃, C₅×S₃², S₃×D₁₅, D₁₅⋊S₃, S₃²×D₅
Quotients: C₁, C₂, C₂², S₃, C₂³, D₅, D₆, D₁₀, C₂²×S₃, S₃², C₂²×D₅, S₃×D₅, C₂×S₃², C₂×S₃×D₅, S₃²×D₅

Permutation representations of S₃²×D₅
►On 30 points - transitive group 30T84

Generators in S₃₀

(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)

(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)

(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)

(1 19)(2 20)(3 16)(4 17)(5 18)(6 26)(7 27)(8 28)(9 29)(10 30)(11 21)(12 22)(13 23)(14 24)(15 25)

(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)

(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)

G:=sub<Sym(30)| (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,19)(2,20)(3,16)(4,17)(5,18)(6,26)(7,27)(8,28)(9,29)(10,30)(11,21)(12,22)(13,23)(14,24)(15,25), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)>;

G:=Group( (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,19)(2,20)(3,16)(4,17)(5,18)(6,26)(7,27)(8,28)(9,29)(10,30)(11,21)(12,22)(13,23)(14,24)(15,25), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29) );

G=PermutationGroup([[(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,26),(7,27),(8,28),(9,29),(10,30),(11,21),(12,22),(13,23),(14,24),(15,25)], [(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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29)]])

G:=TransitiveGroup(30,84);

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	5A	5B	6A	6B	6C	6D	6E	6F	6G	10A	10B	10C	10D	10E	10F	15A	15B	15C	15D	15E	15F	30A	30B	30C	30D
order	1	2	2	2	2	2	2	2	3	3	3	5	5	6	6	6	6	6	6	6	10	10	10	10	10	10	15	15	15	15	15	15	30	30	30	30
size	1	3	3	5	9	15	15	45	2	2	4	2	2	6	6	10	10	20	30	30	6	6	6	6	18	18	4	4	4	4	8	8	12	12	12	12

36 irreducible representations

dim

type

image

C₁

C₂

S₃

D₅

D₆

D₁₀

S₃²

S₃×D₅

C₂×S₃²

C₂×S₃×D₅

S₃²×D₅

kernel

S₃²×D₅

C₃×S₃×D₅

D₅×C₃⋊S₃

C₅×S₃²

S₃×D₁₅

D₁₅⋊S₃

S₃×D₅

S₃²

C₅×S₃

C₃×D₅

D₁₅

C₃×S₃

C₃⋊S₃

D₅

S₃

C₅

C₃

C₁

# reps

Matrix representation of S₃²×D₅ ►in GL₆(𝔽₃₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	30	1
0	0	0	0	30	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	30	0	0	0
0	0	0	30	0	0
0	0	0	0	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	30	1	0	0
0	0	30	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

30	0	0	0	0	0
0	30	0	0	0	0
0	0	0	30	0	0
0	0	30	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

13	1	0	0	0	0
17	30	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	1	0	0	0	0
0	30	0	0	0	0
0	0	30	0	0	0
0	0	0	30	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(31))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,30,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,30,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,30,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,17,0,0,0,0,1,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,1,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S₃²×D₅ in GAP, Magma, Sage, TeX

S_3^2\times D_5

% in TeX

G:=Group("S3^2xD5");

// GroupNames label

G:=SmallGroup(360,137);

// by ID

G=gap.SmallGroup(360,137);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,111,730,10373]);

// Polycyclic

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

// generators/relations

G = S32×D5 order 360 = 23·32·5

Direct product of S3, S3 and D5

G = S₃²×D₅ order 360 = 2³·3²·5

Direct product of S₃, S₃ and D₅