He3.3S3 - GroupNames

G = He₃.₃S₃ order 162 = 2·3⁴

3^rd non-split extension by He₃ of S₃ acting faithfully

non-abelian, supersoluble, monomial

Aliases: He₃.₃S₃, 3_-¹⁺²⋊₂S₃, (C₃×C₉)⋊₅S₃, He₃.C₃⋊₁C₂, C₃².₂(C₃⋊S₃), C₃.₃(He₃⋊C₂), SmallGroup(162,20)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — He₃.C₃ — He₃.₃S₃

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — He₃.C₃ — He₃.₃S₃

Lower central

He₃.C₃ — He₃.₃S₃

Upper central

C₁

Generators and relations for He₃.₃S₃
G = < a,b,c,d,e | a³=b³=c³=e²=1, d³=ebe=b^-1, ab=ba, cac^-1=eae=ab^-1, ad=da, bc=cb, bd=db, dcd^-1=a^-1bc, ece=c^-1, ede=bd² >

C₁

₂₇C₂

C₃

₃C₃

₉C₃

₉S₃

₂₇S₃

₂₇C₆

C₃²

₃C₉

₃C₃²

₃C₉

₃D₉

₃C₃⋊S₃

₃D₉

₉C₃×S₃

3_-¹⁺²

C₃×C₉

He₃

3_-¹⁺²

₃C₉⋊C₆

₃C₃×D₉

₃C₃²⋊C₆

₃C₉⋊C₆

He₃.C₃

He₃.₃S₃

Character table of He₃.₃S₃

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

Permutation representations of He₃.₃S₃
►On 27 points - transitive group 27T42

Generators in S₂₇

(1 4 7)(2 5 8)(3 6 9)(19 25 22)(20 26 23)(21 27 24)

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

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

(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)

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

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

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

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

G:=TransitiveGroup(27,42);

►On 27 points - transitive group 27T68

Generators in S₂₇

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

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

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

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

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

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

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

G:=TransitiveGroup(27,68);

He₃.₃S₃ is a maximal subgroup of
He₃.D₆ He₃.C₃⋊₂C₆ He₃.(C₃⋊S₃) C₃≀C₃⋊S₃
He₃.₃S₃ is a maximal quotient of
He₃.₃Dic₃ (C₃×He₃).S₃ C₃³.(C₃⋊S₃) C₃²⋊C₉⋊₆S₃ C₃.(He₃⋊S₃) C₃²⋊C₉.₁₀S₃ (C₃×C₉)⋊₅D₉ He₃⋊₂D₉ 3_-¹⁺²⋊D₉ He₃.(C₃⋊S₃)

Matrix representation of He₃.₃S₃ ►in GL₆(𝔽₁₉)

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
0	1	0	0	0	0

0	1	0	0	0	0
18	18	0	0	0	0
0	0	0	1	0	0
0	0	18	18	0	0
0	0	0	0	0	1
0	0	0	0	18	18

1	0	0	0	0	0
0	1	0	0	0	0
0	0	18	18	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	18	18

18	15	18	15	4	3
4	3	4	3	16	1
4	3	18	15	18	15
16	1	4	3	4	3
18	15	4	3	18	15
4	3	16	1	4	3

1	0	0	0	0	0
18	18	0	0	0	0
0	0	18	18	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(19))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[18,4,4,16,18,4,15,3,3,1,15,3,18,4,18,4,4,16,15,3,15,3,3,1,4,16,18,4,18,4,3,1,15,3,15,3],[1,18,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,18,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

He₃.₃S₃ in GAP, Magma, Sage, TeX

{\rm He}_3._3S_3

% in TeX

G:=Group("He3.3S3");

// GroupNames label

G:=SmallGroup(162,20);

// by ID

G=gap.SmallGroup(162,20);

# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,41,1802,187,147,2523,728,2704]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of He₃.₃S₃ in TeX
Character table of He₃.₃S₃ in TeX

G = He3.3S3 order 162 = 2·34

3rd non-split extension by He3 of S3 acting faithfully

G = He₃.₃S₃ order 162 = 2·3⁴

3^rd non-split extension by He₃ of S₃ acting faithfully