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G = He3.4S3  order 162 = 2·34

The non-split extension by He3 of S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: He3.4S3, 3- 1+2⋊4S3, C9.(C3×S3), C9⋊S3⋊5C3, (C3×C9)⋊6C6, (C3×C9)⋊7S3, C9○He3⋊1C2, C32.9(C3×S3), C32.6(C3⋊S3), C3.4(C3×C3⋊S3), SmallGroup(162,43)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C9 — He3.4S3
Chief series
C1 — C3 — C32 — C3×C9 — C9○He3 — He3.4S3
Lower central
C3×C9 — He3.4S3
Upper central
C1

Generators and relations for He3.4S3
 G = < a,b,c,d,e | a3=b3=c3=e2=1, d3=b, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ce=ec, ede=b-1d2 >

C1
27C2
C3
3C3
3C3
6C3
9S3
27C6
27S3
C9
C32
C9
C32
C9
2C32
2C9
2C9
2C9
3D9
3C3⋊S3
3D9
3D9
9C3×S3
C3×C9
3- 1+2
3- 1+2
He3
C3×C9
2C3×C9
23- 1+2
23- 1+2
23- 1+2
C9⋊S3
3C9⋊C6
3C9⋊C6
3C3×D9
3C32⋊C6
C9○He3
He3.4S3

Character table of He3.4S3

 class 123A3B3C3D3E3F6A6B9A9B9C9D9E9F9G9H9I9J9K
 size 127233666272722266666666
ρ1111111111111111111111    trivial
ρ21-1111111-1-111111111111    linear of order 2
ρ3111ζ3ζ321ζ3ζ32ζ3ζ32111ζ31ζ32ζ3ζ31ζ32ζ32    linear of order 3
ρ41-11ζ32ζ31ζ32ζ3ζ6ζ65111ζ321ζ3ζ32ζ321ζ3ζ3    linear of order 6
ρ5111ζ32ζ31ζ32ζ3ζ32ζ3111ζ321ζ3ζ32ζ321ζ3ζ3    linear of order 3
ρ61-11ζ3ζ321ζ3ζ32ζ65ζ6111ζ31ζ32ζ3ζ31ζ32ζ32    linear of order 6
ρ720222-1-1-100-1-1-1-1-1-1-1222-1    orthogonal lifted from S3
ρ820222-1-1-1002222-12-1-1-1-1-1    orthogonal lifted from S3
ρ920222-1-1-100-1-1-1-12-12-1-1-12    orthogonal lifted from S3
ρ102022222200-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ11202-1-√-3-1+√-32-1-√-3-1+√-300-1-1-1ζ6-1ζ65ζ6ζ6-1ζ65ζ65    complex lifted from C3×S3
ρ12202-1-√-3-1+√-3-1ζ6ζ6500-1-1-1ζ62ζ65-1-√-3ζ6-1ζ65-1+√-3    complex lifted from C3×S3
ρ13202-1+√-3-1-√-3-1ζ65ζ600-1-1-1ζ65-1ζ6ζ65-1+√-32-1-√-3ζ6    complex lifted from C3×S3
ρ14202-1-√-3-1+√-3-1ζ6ζ6500222-1-√-3-1-1+√-3ζ6ζ6-1ζ65ζ65    complex lifted from C3×S3
ρ15202-1+√-3-1-√-32-1+√-3-1-√-300-1-1-1ζ65-1ζ6ζ65ζ65-1ζ6ζ6    complex lifted from C3×S3
ρ16202-1-√-3-1+√-3-1ζ6ζ6500-1-1-1ζ6-1ζ65ζ6-1-√-32-1+√-3ζ65    complex lifted from C3×S3
ρ17202-1+√-3-1-√-3-1ζ65ζ600-1-1-1ζ652ζ6-1+√-3ζ65-1ζ6-1-√-3    complex lifted from C3×S3
ρ18202-1+√-3-1-√-3-1ζ65ζ600222-1+√-3-1-1-√-3ζ65ζ65-1ζ6ζ6    complex lifted from C3×S3
ρ1960-300000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ9400000000    orthogonal faithful
ρ2060-300000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ9200000000    orthogonal faithful
ρ2160-300000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ900000000    orthogonal faithful

Permutation representations of He3.4S3
►On 27 points - transitive group 27T54
Generators in S27
(1 14 26)(2 15 27)(3 16 19)(4 17 20)(5 18 21)(6 10 22)(7 11 23)(8 12 24)(9 13 25)

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

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

(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 21)(11 20)(12 19)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)
 
G:=sub<Sym(27)| (1,14,26)(2,15,27)(3,16,19)(4,17,20)(5,18,21)(6,10,22)(7,11,23)(8,12,24)(9,13,25), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (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)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)>;
 
G:=Group( (1,14,26)(2,15,27)(3,16,19)(4,17,20)(5,18,21)(6,10,22)(7,11,23)(8,12,24)(9,13,25), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (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)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22) );
 
G=PermutationGroup([[(1,14,26),(2,15,27),(3,16,19),(4,17,20),(5,18,21),(6,10,22),(7,11,23),(8,12,24),(9,13,25)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27)], [(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24)], [(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),(10,21),(11,20),(12,19),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22)]])
 
G:=TransitiveGroup(27,54);
 

►On 27 points - transitive group 27T56
Generators in S27
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)

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

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

(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 16)(11 15)(12 14)(17 18)(19 25)(20 24)(21 23)(26 27)
 
G:=sub<Sym(27)| (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (1,22,13)(2,23,14)(3,24,15)(4,25,16)(5,26,17)(6,27,18)(7,19,10)(8,20,11)(9,21,12), (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)(10,16)(11,15)(12,14)(17,18)(19,25)(20,24)(21,23)(26,27)>;
 
G:=Group( (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (1,22,13)(2,23,14)(3,24,15)(4,25,16)(5,26,17)(6,27,18)(7,19,10)(8,20,11)(9,21,12), (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)(10,16)(11,15)(12,14)(17,18)(19,25)(20,24)(21,23)(26,27) );
 
G=PermutationGroup([[(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27)], [(1,22,13),(2,23,14),(3,24,15),(4,25,16),(5,26,17),(6,27,18),(7,19,10),(8,20,11),(9,21,12)], [(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),(10,16),(11,15),(12,14),(17,18),(19,25),(20,24),(21,23),(26,27)]])
 
G:=TransitiveGroup(27,56);
 

He3.4S3 is a maximal subgroup of
 He3.6D6  He3.D9  He3.2D9  He3.3D9  He3.4D9  He3.5D9  C3≀C3.S3  C3≀C3⋊S3  3- 1+4⋊C2  C9○He3⋊3S3
He3.4S3 is a maximal quotient of
 He3.4Dic3  C92⋊3S3  C92⋊3C6  He3⋊3D9  C92⋊9C6  C9⋊He3⋊2C2  (C32×C9)⋊S3  (C32×C9)⋊C6  C92⋊6S3  C92⋊10C6  C92⋊4C6  C92⋊5S3  C92⋊5C6  C92⋊11C6  C9○He3⋊3S3

Matrix representation of He3.4S3 ►in GL6(𝔽19)

100000
010000
000100
00181800
00001818
000010
,
010000
18180000
000100
00181800
000001
00001818
,
001000
000100
000010
000001
100000
010000
,
1420000
17120000
0014200
00171200
0000142
00001712
,
100000
18180000
001000
00181800
000010
00001818

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[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],[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],[14,17,0,0,0,0,2,12,0,0,0,0,0,0,14,17,0,0,0,0,2,12,0,0,0,0,0,0,14,17,0,0,0,0,2,12],[1,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,18] >;
 

He3.4S3 in GAP, Magma, Sage, TeX 

{\rm He}_3._4S_3
 
% in TeX
 
G:=Group("He3.4S3");
 
// GroupNames label
 
G:=SmallGroup(162,43);
 
// by ID
 
G=gap.SmallGroup(162,43);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-3,-3,992,282,723,728,2704]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^3=b,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=b^-1*d^2>;
 
// generators/relations
 

Export

Subgroup lattice of He3.4S3 in TeX
Character table of He3.4S3 in TeX

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