He3:5S3 - GroupNames

G = He₃⋊₅S₃ order 162 = 2·3⁴

2^nd semidirect product of He₃ and S₃ acting via S₃/C₃=C₂

non-abelian, supersoluble, monomial

Aliases: He₃⋊₅S₃, C₃³⋊₆S₃, (C₃×He₃)⋊₅C₂, C₃⋊(He₃⋊C₂), C₃²⋊₂(C₃⋊S₃), C₃.₂(C₃³⋊C₂), SmallGroup(162,46)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₃×He₃ — He₃⋊₅S₃

Chief series

C₁ — C₃ — C₃² — He₃ — C₃×He₃ — He₃⋊₅S₃

Lower central

C₃×He₃ — He₃⋊₅S₃

Upper central

C₁ — C₃

Generators and relations for He₃⋊₅S₃
G = < a,b,c,d,e | a³=b³=c³=d³=e²=1, dad^-1=ab=ba, cac^-1=ab^-1, eae=a^-1, bc=cb, bd=db, be=eb, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 444 in 97 conjugacy classes, 31 normal (6 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₃², C₃², C₃², C₃×S₃, C₃⋊S₃, He₃, C₃³, He₃⋊C₂, C₃×C₃⋊S₃, C₃×He₃, He₃⋊₅S₃
Quotients: C₁, C₂, S₃, C₃⋊S₃, He₃⋊C₂, C₃³⋊C₂, He₃⋊₅S₃

Character table of He₃⋊₅S₃

class	1	2	3A	3B	3C	3D	3E	3F	3G	3H	3I	3J	3K	3L	3M	3N	3O	3P	3Q	6A	6B
size	1	27	1	1	2	2	2	6	6	6	6	6	6	6	6	6	6	6	6	27	27
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	linear of order 2
ρ₃	2	0	2	2	2	2	2	-1	2	-1	-1	-1	2	-1	-1	2	-1	-1	-1	0	0	orthogonal lifted from S₃
ρ₄	2	0	2	2	2	2	2	-1	-1	-1	2	-1	-1	-1	2	-1	2	-1	-1	0	0	orthogonal lifted from S₃
ρ₅	2	0	2	2	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	2	2	2	2	0	0	orthogonal lifted from S₃
ρ₆	2	0	2	2	-1	-1	-1	-1	2	2	2	-1	-1	-1	-1	-1	-1	-1	2	0	0	orthogonal lifted from S₃
ρ₇	2	0	2	2	-1	-1	-1	-1	2	-1	-1	2	-1	-1	2	-1	-1	2	-1	0	0	orthogonal lifted from S₃
ρ₈	2	0	2	2	-1	-1	-1	-1	-1	2	-1	2	2	-1	-1	-1	2	-1	-1	0	0	orthogonal lifted from S₃
ρ₉	2	0	2	2	-1	-1	-1	-1	-1	-1	-1	-1	2	2	2	-1	-1	-1	2	0	0	orthogonal lifted from S₃
ρ₁₀	2	0	2	2	-1	-1	-1	-1	-1	-1	2	2	-1	2	-1	2	-1	-1	-1	0	0	orthogonal lifted from S₃
ρ₁₁	2	0	2	2	-1	-1	-1	2	-1	-1	2	-1	2	-1	-1	-1	-1	2	-1	0	0	orthogonal lifted from S₃
ρ₁₂	2	0	2	2	-1	-1	-1	2	2	-1	-1	-1	-1	2	-1	-1	2	-1	-1	0	0	orthogonal lifted from S₃
ρ₁₃	2	0	2	2	-1	-1	-1	2	-1	2	-1	-1	-1	-1	2	2	-1	-1	-1	0	0	orthogonal lifted from S₃
ρ₁₄	2	0	2	2	2	2	2	-1	-1	2	-1	-1	-1	2	-1	-1	-1	2	-1	0	0	orthogonal lifted from S₃
ρ₁₅	2	0	2	2	2	2	2	2	-1	-1	-1	2	-1	-1	-1	-1	-1	-1	2	0	0	orthogonal lifted from S₃
ρ₁₆	3	1	^-3+3√-3/₂	^-3-3√-3/₂	^-3-3√-3/₂	^-3+3√-3/₂	3	0	0	0	0	0	0	0	0	0	0	0	0	ζ₃²	ζ₃	complex lifted from He₃⋊C₂
ρ₁₇	3	1	^-3-3√-3/₂	^-3+3√-3/₂	^-3+3√-3/₂	^-3-3√-3/₂	3	0	0	0	0	0	0	0	0	0	0	0	0	ζ₃	ζ₃²	complex lifted from He₃⋊C₂
ρ₁₈	3	-1	^-3-3√-3/₂	^-3+3√-3/₂	^-3+3√-3/₂	^-3-3√-3/₂	3	0	0	0	0	0	0	0	0	0	0	0	0	ζ₆⁵	ζ₆	complex lifted from He₃⋊C₂
ρ₁₉	3	-1	^-3+3√-3/₂	^-3-3√-3/₂	^-3-3√-3/₂	^-3+3√-3/₂	3	0	0	0	0	0	0	0	0	0	0	0	0	ζ₆	ζ₆⁵	complex lifted from He₃⋊C₂
ρ₂₀	6	0	-3-3√-3	-3+3√-3	^3-3√-3/₂	^3+3√-3/₂	-3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful
ρ₂₁	6	0	-3+3√-3	-3-3√-3	^3+3√-3/₂	^3-3√-3/₂	-3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful

Permutation representations of He₃⋊₅S₃
►On 18 points - transitive group 18T89

Generators in S₁₈

(7 8 9)(10 11 12)(13 14 15)(16 17 18)

(1 6 5)(2 4 3)(7 9 8)(10 12 11)(13 14 15)(16 17 18)

(1 9 14)(2 10 18)(3 11 17)(4 12 16)(5 7 13)(6 8 15)

(1 15 7)(2 17 12)(3 16 10)(4 18 11)(5 14 8)(6 13 9)

(1 3)(2 6)(4 5)(7 16)(8 18)(9 17)(10 15)(11 14)(12 13)

G:=sub<Sym(18)| (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,6,5)(2,4,3)(7,9,8)(10,12,11)(13,14,15)(16,17,18), (1,9,14)(2,10,18)(3,11,17)(4,12,16)(5,7,13)(6,8,15), (1,15,7)(2,17,12)(3,16,10)(4,18,11)(5,14,8)(6,13,9), (1,3)(2,6)(4,5)(7,16)(8,18)(9,17)(10,15)(11,14)(12,13)>;

G:=Group( (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,6,5)(2,4,3)(7,9,8)(10,12,11)(13,14,15)(16,17,18), (1,9,14)(2,10,18)(3,11,17)(4,12,16)(5,7,13)(6,8,15), (1,15,7)(2,17,12)(3,16,10)(4,18,11)(5,14,8)(6,13,9), (1,3)(2,6)(4,5)(7,16)(8,18)(9,17)(10,15)(11,14)(12,13) );

G=PermutationGroup([[(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,6,5),(2,4,3),(7,9,8),(10,12,11),(13,14,15),(16,17,18)], [(1,9,14),(2,10,18),(3,11,17),(4,12,16),(5,7,13),(6,8,15)], [(1,15,7),(2,17,12),(3,16,10),(4,18,11),(5,14,8),(6,13,9)], [(1,3),(2,6),(4,5),(7,16),(8,18),(9,17),(10,15),(11,14),(12,13)]])

G:=TransitiveGroup(18,89);

►On 27 points - transitive group 27T74

Generators in S₂₇

(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)

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

(2 14 12)(3 10 15)(4 8 26)(6 25 7)(16 19 22)(17 23 20)

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

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

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

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

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

G:=TransitiveGroup(27,74);

He₃⋊₅S₃ is a maximal subgroup of
He₃⋊₅D₆ S₃×He₃⋊C₂ C₃.C₃≀S₃ C₃²⋊C₉⋊S₃ (C₃×He₃).C₆ C₃⁴⋊₃S₃ (C₃²×C₉)⋊₈S₃ C₃⁴⋊₅S₃ He₃.C₃⋊S₃ He₃⋊C₃⋊₂S₃ He₃⋊₄D₉ C₉○He₃⋊₄S₃ C₃⁴⋊₁₃S₃ 3₊¹⁺⁴⋊₃C₂
He₃⋊₅S₃ is a maximal quotient of
He₃⋊₆Dic₃ C₃³⋊₆D₉ He₃⋊₄D₉ C₃⁴⋊₆S₃ C₃⁴⋊₇S₃ He₃.(C₃⋊S₃) C₃⋊(He₃⋊S₃) (C₃²×C₉).S₃ C₃≀C₃⋊S₃ C₃⁴⋊₁₃S₃

Matrix representation of He₃⋊₅S₃ ►in GL₅(𝔽₇)

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	2	0
0	0	0	0	4

1	0	0	0	0
0	1	0	0	0
0	0	4	0	0
0	0	0	4	0
0	0	0	0	4

1	0	0	0	0
0	1	0	0	0
0	0	0	0	1
0	0	1	0	0
0	0	0	1	0

6	6	0	0	0
1	0	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

1	0	0	0	0
6	6	0	0	0
0	0	6	0	0
0	0	0	0	6
0	0	0	6	0

G:=sub<GL(5,GF(7))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[6,1,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,6,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,6,0] >;

He₃⋊₅S₃ in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_5S_3

% in TeX

G:=Group("He3:5S3");

// GroupNames label

G:=SmallGroup(162,46);

// by ID

G=gap.SmallGroup(162,46);

# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,41,182,723,253]);

// Polycyclic

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

// generators/relations

Export

Character table of He₃⋊₅S₃ in TeX

G = He3⋊5S3 order 162 = 2·34

2nd semidirect product of He3 and S3 acting via S3/C3=C2

G = He₃⋊₅S₃ order 162 = 2·3⁴

2^nd semidirect product of He₃ and S₃ acting via S₃/C₃=C₂