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G = He3.2C6order 162 = 2·34

2nd non-split extension by He3 of C6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He3.2C6, (C3×C9)⋊3S3, He3⋊C31C2, He3⋊C22C3, C32.3(C3×S3), C3.8(C32⋊C6), SmallGroup(162,14)

Series: Derived Chief Lower central Upper central

C1C3He3 — He3.2C6
C1C3C32He3He3⋊C3 — He3.2C6
He3 — He3.2C6
C1C3

Generators and relations for He3.2C6
 G = < a,b,c,d | a3=b3=c3=1, d6=b, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a-1c-1 >

9C2
3C3
9C3
18C3
3S3
9C6
9S3
3C32
3C9
6C32
3C3×S3
9C18
9C3×S3
2He3
3S3×C9

Character table of He3.2C6

 class 123A3B3C3D3E3F6A6B9A9B9C9D9E9F18A18B18C18D18E18F
 size 1911618181899333333999999
ρ11111111111111111111111    trivial
ρ21-1111111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ311111ζ31ζ3211ζ3ζ32ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ41-1111ζ31ζ32-1-1ζ3ζ32ζ32ζ32ζ3ζ3ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ51-1111ζ321ζ3-1-1ζ32ζ3ζ3ζ3ζ32ζ32ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ611111ζ321ζ311ζ32ζ3ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ720222-1-1-100222222000000    orthogonal lifted from S3
ρ820222ζ6-1ζ6500-1--3-1+-3-1+-3-1+-3-1--3-1--3000000    complex lifted from C3×S3
ρ920222ζ65-1ζ600-1+-3-1--3-1--3-1--3-1+-3-1+-3000000    complex lifted from C3×S3
ρ103-1-3-3-3/2-3+3-3/20000ζ6ζ6598929499794ζ97+2ζ9ζ95+2ζ92ζ98+2ζ9597994959892    complex faithful
ρ1131-3-3-3/2-3+3-3/20000ζ32ζ3ζ98+2ζ959794ζ97+2ζ99499892ζ95+2ζ92ζ9ζ94ζ97ζ92ζ95ζ98    complex faithful
ρ123-1-3-3-3/2-3+3-3/20000ζ6ζ65ζ98+2ζ959794ζ97+2ζ99499892ζ95+2ζ9299497929598    complex faithful
ρ133-1-3+3-3/2-3-3-3/20000ζ65ζ6949ζ95+2ζ929892ζ98+2ζ95ζ97+2ζ9979498959297949    complex faithful
ρ143-1-3+3-3/2-3-3-3/20000ζ65ζ697949892ζ98+2ζ95ζ95+2ζ92949ζ97+2ζ995929899794    complex faithful
ρ153-1-3-3-3/2-3+3-3/20000ζ6ζ65ζ95+2ζ92ζ97+2ζ99499794ζ98+2ζ95989294979989295    complex faithful
ρ163-1-3+3-3/2-3-3-3/20000ζ65ζ6ζ97+2ζ9ζ98+2ζ95ζ95+2ζ929892979494992989594997    complex faithful
ρ1731-3-3-3/2-3+3-3/20000ζ32ζ398929499794ζ97+2ζ9ζ95+2ζ92ζ98+2ζ95ζ97ζ9ζ94ζ95ζ98ζ92    complex faithful
ρ1831-3+3-3/2-3-3-3/20000ζ3ζ32ζ97+2ζ9ζ98+2ζ95ζ95+2ζ9298929794949ζ92ζ98ζ95ζ94ζ9ζ97    complex faithful
ρ1931-3-3-3/2-3+3-3/20000ζ32ζ3ζ95+2ζ92ζ97+2ζ99499794ζ98+2ζ959892ζ94ζ97ζ9ζ98ζ92ζ95    complex faithful
ρ2031-3+3-3/2-3-3-3/20000ζ3ζ3297949892ζ98+2ζ95ζ95+2ζ92949ζ97+2ζ9ζ95ζ92ζ98ζ9ζ97ζ94    complex faithful
ρ2131-3+3-3/2-3-3-3/20000ζ3ζ32949ζ95+2ζ929892ζ98+2ζ95ζ97+2ζ99794ζ98ζ95ζ92ζ97ζ94ζ9    complex faithful
ρ226066-300000000000000000    orthogonal lifted from C32⋊C6

Permutation representations of He3.2C6
On 27 points - transitive group 27T40
Generators in S27
(1 4 7)(2 5 8)(3 6 9)(10 16 22)(12 18 24)(14 20 26)
(1 7 4)(2 8 5)(3 9 6)(10 16 22)(11 17 23)(12 18 24)(13 19 25)(14 20 26)(15 21 27)
(1 24 15)(2 22 13)(3 20 11)(4 18 27)(5 16 25)(6 14 23)(7 12 21)(8 10 19)(9 26 17)
(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)

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

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

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

G:=TransitiveGroup(27,40);

On 27 points - transitive group 27T49
Generators in S27
(1 24 27)(2 16 19)(3 26 11)(4 18 21)(5 10 13)(6 20 23)(7 12 15)(8 22 25)(9 14 17)
(1 7 4)(2 8 5)(3 9 6)(10 16 22)(11 17 23)(12 18 24)(13 19 25)(14 20 26)(15 21 27)
(1 24 15)(2 19 10)(4 18 27)(5 13 22)(7 12 21)(8 25 16)(11 23 17)(14 20 26)
(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)

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

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

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

G:=TransitiveGroup(27,49);

He3.2C6 is a maximal subgroup of
He3.2D6  C922S3  C3≀S33C3  He3.(C3×C6)  C3≀C3.C6  He3⋊C32S3
He3.2C6 is a maximal quotient of
He3.2C12  C32⋊C9.S3  (C3×He3).C6  C32⋊C9.C6  (C3×C9)⋊3D9  He3⋊C18  C92⋊S3  C92.S3  C9⋊C9.S3  C9⋊C9.3S3  C9⋊C9⋊S3  He3⋊C32S3

Matrix representation of He3.2C6 in GL3(𝔽19) generated by

010
001
100
,
700
070
007
,
0110
007
100
,
131013
131515
101015
G:=sub<GL(3,GF(19))| [0,0,1,1,0,0,0,1,0],[7,0,0,0,7,0,0,0,7],[0,0,1,11,0,0,0,7,0],[13,13,10,10,15,10,13,15,15] >;

He3.2C6 in GAP, Magma, Sage, TeX

{\rm He}_3._2C_6
% in TeX

G:=Group("He3.2C6");
// GroupNames label

G:=SmallGroup(162,14);
// by ID

G=gap.SmallGroup(162,14);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,276,182,187,1803,253]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=1,d^6=b,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*c^-1>;
// generators/relations

Export

Subgroup lattice of He3.2C6 in TeX
Character table of He3.2C6 in TeX

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