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G = He3.4C6order 162 = 2·34

The non-split extension by He3 of C6 acting via C6/C3=C2

non-abelian, supersoluble, monomial

Aliases: He3.4C6, (C3×C9)⋊8S3, C9○He32C2, C9.2(C3⋊S3), C32.5(C3×S3), He3⋊C2.2C3, C3.7(C3×C3⋊S3), SmallGroup(162,44)

Series: Derived Chief Lower central Upper central

C1C3He3 — He3.4C6
C1C3C32He3C9○He3 — He3.4C6
He3 — He3.4C6
C1C9

Generators and relations for He3.4C6
 G = < a,b,c,d | a3=b3=c3=1, d6=b, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >

9C2
3C3
3C3
3C3
3C3
3S3
3S3
3S3
3S3
9C6
2C9
2C9
2C9
2C9
3C3×S3
3C3×S3
3C3×S3
3C3×S3
9C18
23- 1+2
23- 1+2
23- 1+2
23- 1+2
3S3×C9
3S3×C9
3S3×C9
3S3×C9

Character table of He3.4C6

 class 123A3B3C3D3E3F6A6B9A9B9C9D9E9F9G9H9I9J9K9L9M9N18A18B18C18D18E18F
 size 191166669911111166666666999999
ρ1111111111111111111111111111111    trivial
ρ21-1111111-1-111111111111111-1-1-1-1-1-1    linear of order 2
ρ31-1111111-1-1ζ32ζ3ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ41111111111ζ3ζ32ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ51111111111ζ32ζ3ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ61-1111111-1-1ζ3ζ32ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ72022-1-12-100222222-1-1-1-122-1-1000000    orthogonal lifted from S3
ρ820222-1-1-100222222-1-12-1-1-1-12000000    orthogonal lifted from S3
ρ92022-1-1-1200222222-12-1-1-1-12-1000000    orthogonal lifted from S3
ρ102022-12-1-1002222222-1-12-1-1-1-1000000    orthogonal lifted from S3
ρ112022-1-12-100-1+-3-1--3-1+-3-1--3-1--3-1+-3ζ6ζ65ζ65ζ65-1+-3-1--3ζ6ζ6000000    complex lifted from C3×S3
ρ122022-1-1-1200-1--3-1+-3-1--3-1+-3-1+-3-1--3ζ65-1--3ζ6ζ6ζ6ζ65-1+-3ζ65000000    complex lifted from C3×S3
ρ132022-1-1-1200-1+-3-1--3-1+-3-1--3-1--3-1+-3ζ6-1+-3ζ65ζ65ζ65ζ6-1--3ζ6000000    complex lifted from C3×S3
ρ142022-12-1-100-1+-3-1--3-1+-3-1--3-1--3-1+-3-1--3ζ65ζ65-1+-3ζ65ζ6ζ6ζ6000000    complex lifted from C3×S3
ρ1520222-1-1-100-1--3-1+-3-1--3-1+-3-1+-3-1--3ζ65ζ6-1--3ζ6ζ6ζ65ζ65-1+-3000000    complex lifted from C3×S3
ρ162022-1-12-100-1--3-1+-3-1--3-1+-3-1+-3-1--3ζ65ζ6ζ6ζ6-1--3-1+-3ζ65ζ65000000    complex lifted from C3×S3
ρ172022-12-1-100-1--3-1+-3-1--3-1+-3-1+-3-1--3-1+-3ζ6ζ6-1--3ζ6ζ65ζ65ζ65000000    complex lifted from C3×S3
ρ1820222-1-1-100-1+-3-1--3-1+-3-1--3-1--3-1+-3ζ6ζ65-1+-3ζ65ζ65ζ6ζ6-1--3000000    complex lifted from C3×S3
ρ193-1-3-3-3/2-3+3-3/20000ζ6ζ65929495979980000000094979989295    complex faithful
ρ2031-3-3-3/2-3+3-3/20000ζ32ζ39599894979200000000ζ9ζ94ζ97ζ92ζ95ζ98    complex faithful
ρ213-1-3+3-3/2-3-3-3/20000ζ65ζ6979594929890000000095929899794    complex faithful
ρ2231-3+3-3/2-3-3-3/20000ζ3ζ329795949298900000000ζ95ζ92ζ98ζ9ζ97ζ94    complex faithful
ρ233-1-3-3-3/2-3+3-3/20000ζ6ζ65959989497920000000099497929598    complex faithful
ρ2431-3-3-3/2-3+3-3/20000ζ32ζ39897929949500000000ζ97ζ9ζ94ζ95ζ98ζ92    complex faithful
ρ2531-3+3-3/2-3-3-3/20000ζ3ζ329929798959400000000ζ92ζ98ζ95ζ94ζ9ζ97    complex faithful
ρ263-1-3-3-3/2-3+3-3/20000ζ6ζ65989792994950000000097994959892    complex faithful
ρ273-1-3+3-3/2-3-3-3/20000ζ65ζ6992979895940000000092989594997    complex faithful
ρ2831-3+3-3/2-3-3-3/20000ζ3ζ329498995929700000000ζ98ζ95ζ92ζ97ζ94ζ9    complex faithful
ρ2931-3-3-3/2-3+3-3/20000ζ32ζ39294959799800000000ζ94ζ97ζ9ζ98ζ92ζ95    complex faithful
ρ303-1-3+3-3/2-3-3-3/20000ζ65ζ6949899592970000000098959297949    complex faithful

Permutation representations of He3.4C6
On 27 points - transitive group 27T39
Generators in S27
(1 13 22)(2 23 14)(3 15 24)(4 25 16)(5 17 26)(6 27 18)(7 19 10)(8 11 20)(9 21 12)
(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)
(10 22 16)(11 17 23)(12 24 18)(13 19 25)(14 26 20)(15 21 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)

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

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

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

G:=TransitiveGroup(27,39);

He3.4C6 is a maximal subgroup of
He3.3C12  He3.6D6  He3.C18  He3.2C18  C3≀S33C3  C3≀C3.C6  He3.5C18  3- 1+42C2  C9○He34S3
He3.4C6 is a maximal quotient of
He3.5C12  C924S3  C9×He3⋊C2  (C32×C9)⋊8S3  C9⋊C92S3  C926S3  C925S3  C9○He34S3

Matrix representation of He3.4C6 in GL3(𝔽19) generated by

100
0110
007
,
1100
0110
0011
,
010
001
100
,
600
006
060
G:=sub<GL(3,GF(19))| [1,0,0,0,11,0,0,0,7],[11,0,0,0,11,0,0,0,11],[0,0,1,1,0,0,0,1,0],[6,0,0,0,0,6,0,6,0] >;

He3.4C6 in GAP, Magma, Sage, TeX

{\rm He}_3._4C_6
% in TeX

G:=Group("He3.4C6");
// GroupNames label

G:=SmallGroup(162,44);
// by ID

G=gap.SmallGroup(162,44);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,276,182,723,253]);
// 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*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of He3.4C6 in TeX
Character table of He3.4C6 in TeX

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