He3:S3 - GroupNames

G = He₃⋊S₃ order 162 = 2·3⁴

3^rd semidirect product of He₃ and S₃ acting faithfully

non-abelian, supersoluble, monomial

Aliases: He₃⋊₃S₃, (C₃×C₉)⋊₆S₃, He₃⋊C₃⋊₃C₂, C₃².₃(C₃⋊S₃), C₃.₄(He₃⋊C₂), SmallGroup(162,21)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — He₃⋊C₃ — He₃⋊S₃

Chief series

C₁ — C₃ — C₃² — He₃ — He₃⋊C₃ — He₃⋊S₃

Lower central

He₃⋊C₃ — He₃⋊S₃

Upper central

C₁

Generators and relations for He₃⋊S₃
G = < a,b,c,d,e | a³=b³=c³=d³=e²=1, dad^-1=ab=ba, cac^-1=ab^-1, ae=ea, bc=cb, bd=db, ebe=b^-1, dcd^-1=a^-1b^-1c, ece=ab^-1c^-1, ede=d^-1 >

C₁

₂₇C₂

C₃

₃C₃

₉C₃

₉S₃

₂₇S₃

₂₇C₆

₂₇S₃

C₃²

₃C₉

₃C₃²

₃D₉

₃C₃⋊S₃

₉C₃×S₃

He₃

C₃×C₉

₃C₃²⋊C₆

₃C₃×D₉

He₃⋊C₃

He₃⋊S₃

Character table of He₃⋊S₃

class	1	2	3A	3B	3C	3D	3E	3F	6A	6B	9A	9B	9C
size	1	27	2	3	3	18	18	18	27	27	6	6	6
ρ₁	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	linear of order 2
ρ₃	2	0	2	2	2	-1	-1	-1	0	0	2	2	2	orthogonal lifted from S₃
ρ₄	2	0	2	2	2	-1	-1	2	0	0	-1	-1	-1	orthogonal lifted from S₃
ρ₅	2	0	2	2	2	2	-1	-1	0	0	-1	-1	-1	orthogonal lifted from S₃
ρ₆	2	0	2	2	2	-1	2	-1	0	0	-1	-1	-1	orthogonal lifted from S₃
ρ₇	3	1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	ζ₃	ζ₃²	0	0	0	complex lifted from He₃⋊C₂
ρ₈	3	-1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	ζ₆	ζ₆⁵	0	0	0	complex lifted from He₃⋊C₂
ρ₉	3	-1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	ζ₆⁵	ζ₆	0	0	0	complex lifted from He₃⋊C₂
ρ₁₀	3	1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	ζ₃²	ζ₃	0	0	0	complex lifted from He₃⋊C₂
ρ₁₁	6	0	-3	0	0	0	0	0	0	0	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	orthogonal faithful
ρ₁₂	6	0	-3	0	0	0	0	0	0	0	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	orthogonal faithful
ρ₁₃	6	0	-3	0	0	0	0	0	0	0	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	orthogonal faithful

Permutation representations of He₃⋊S₃
►On 27 points - transitive group 27T44

Generators in S₂₇

(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)

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

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

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

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

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

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

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

G:=TransitiveGroup(27,44);

►On 27 points - transitive group 27T66

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

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

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

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

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

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

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

G:=TransitiveGroup(27,66);

He₃⋊S₃ is a maximal subgroup of
He₃.₂D₆ C₉²⋊S₃ C₉⋊C₉⋊S₃ He₃⋊(C₃×S₃) C₃⋊(He₃⋊S₃) C₃≀C₃⋊S₃
He₃⋊S₃ is a maximal quotient of
He₃⋊Dic₃ (C₃×He₃)⋊S₃ C₃²⋊C₉⋊₆S₃ C₃.(He₃⋊S₃) (C₃×C₉)⋊₆D₉ He₃⋊₂D₉ C₉²⋊₂S₃ C₃⋊(He₃⋊S₃)

Matrix representation of He₃⋊S₃ ►in GL₆(𝔽₁₉)

0	0	18	1	0	0
7	7	17	18	0	0
0	0	12	0	1	0
0	0	12	0	0	1
0	0	8	0	0	0
1	0	8	0	0	0

0	18	0	0	0	0
1	18	0	0	0	0
7	0	18	18	0	0
0	12	1	0	0	0
11	0	0	0	18	18
0	8	0	0	1	0

7	5	7	5	7	5
2	7	2	7	2	7
10	16	1	12	6	10
7	11	18	5	1	12
12	1	13	9	11	2
4	0	11	2	18	7

5	3	14	2	17	12
16	2	12	14	5	17
8	14	7	8	4	13
15	0	14	13	11	18
6	14	10	4	10	8
7	3	17	9	8	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	0	0	0
7	7	18	18	0	0
0	0	0	0	1	0
11	11	0	0	18	18

G:=sub<GL(6,GF(19))| [0,7,0,0,0,1,0,7,0,0,0,0,18,17,12,12,8,8,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,7,0,11,0,18,18,0,12,0,8,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[7,2,10,7,12,4,5,7,16,11,1,0,7,2,1,18,13,11,5,7,12,5,9,2,7,2,6,1,11,18,5,7,10,12,2,7],[5,16,8,15,6,7,3,2,14,0,14,3,14,12,7,14,10,17,2,14,8,13,4,9,17,5,4,11,10,8,12,17,13,18,8,1],[0,1,0,7,0,11,1,0,0,7,0,11,0,0,1,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,18] >;

He₃⋊S₃ in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes S_3

% in TeX

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

// GroupNames label

G:=SmallGroup(162,21);

// by ID

G=gap.SmallGroup(162,21);

# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,41,182,187,728,433,2704]);

// 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,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=a^-1*b^-1*c,e*c*e=a*b^-1*c^-1,e*d*e=d^-1>;

// generators/relations

Export

Subgroup lattice of He₃⋊S₃ in TeX
Character table of He₃⋊S₃ in TeX

G = He3⋊S3 order 162 = 2·34

3rd semidirect product of He3 and S3 acting faithfully

G = He₃⋊S₃ order 162 = 2·3⁴

3^rd semidirect product of He₃ and S₃ acting faithfully