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## G = He3.2C6order 162 = 2·34

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

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

 Derived series C1 — C3 — He3 — He3.2C6
 Chief series C1 — C3 — C32 — He3 — He3⋊C3 — He3.2C6
 Lower central He3 — He3.2C6
 Upper central C1 — C3

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 >

Character table of He3.2C6

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 9A 9B 9C 9D 9E 9F 18A 18B 18C 18D 18E 18F size 1 9 1 1 6 18 18 18 9 9 3 3 3 3 3 3 9 9 9 9 9 9 ρ1 1 1 1 1 1 1 1 1 1 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 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 ζ3 1 ζ32 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ4 1 -1 1 1 1 ζ3 1 ζ32 -1 -1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 linear of order 6 ρ5 1 -1 1 1 1 ζ32 1 ζ3 -1 -1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 linear of order 6 ρ6 1 1 1 1 1 ζ32 1 ζ3 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ7 2 0 2 2 2 -1 -1 -1 0 0 2 2 2 2 2 2 0 0 0 0 0 0 orthogonal lifted from S3 ρ8 2 0 2 2 2 ζ6 -1 ζ65 0 0 -1-√-3 -1+√-3 -1+√-3 -1+√-3 -1-√-3 -1-√-3 0 0 0 0 0 0 complex lifted from C3×S3 ρ9 2 0 2 2 2 ζ65 -1 ζ6 0 0 -1+√-3 -1-√-3 -1-√-3 -1-√-3 -1+√-3 -1+√-3 0 0 0 0 0 0 complex lifted from C3×S3 ρ10 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ6 ζ65 2ζ98+ζ92 2ζ94+ζ9 2ζ97+ζ94 ζ97+2ζ9 ζ95+2ζ92 ζ98+2ζ95 -ζ97 -ζ9 -ζ94 -ζ95 -ζ98 -ζ92 complex faithful ρ11 3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ32 ζ3 ζ98+2ζ95 2ζ97+ζ94 ζ97+2ζ9 2ζ94+ζ9 2ζ98+ζ92 ζ95+2ζ92 ζ9 ζ94 ζ97 ζ92 ζ95 ζ98 complex faithful ρ12 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ6 ζ65 ζ98+2ζ95 2ζ97+ζ94 ζ97+2ζ9 2ζ94+ζ9 2ζ98+ζ92 ζ95+2ζ92 -ζ9 -ζ94 -ζ97 -ζ92 -ζ95 -ζ98 complex faithful ρ13 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ65 ζ6 2ζ94+ζ9 ζ95+2ζ92 2ζ98+ζ92 ζ98+2ζ95 ζ97+2ζ9 2ζ97+ζ94 -ζ98 -ζ95 -ζ92 -ζ97 -ζ94 -ζ9 complex faithful ρ14 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ65 ζ6 2ζ97+ζ94 2ζ98+ζ92 ζ98+2ζ95 ζ95+2ζ92 2ζ94+ζ9 ζ97+2ζ9 -ζ95 -ζ92 -ζ98 -ζ9 -ζ97 -ζ94 complex faithful ρ15 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ6 ζ65 ζ95+2ζ92 ζ97+2ζ9 2ζ94+ζ9 2ζ97+ζ94 ζ98+2ζ95 2ζ98+ζ92 -ζ94 -ζ97 -ζ9 -ζ98 -ζ92 -ζ95 complex faithful ρ16 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ65 ζ6 ζ97+2ζ9 ζ98+2ζ95 ζ95+2ζ92 2ζ98+ζ92 2ζ97+ζ94 2ζ94+ζ9 -ζ92 -ζ98 -ζ95 -ζ94 -ζ9 -ζ97 complex faithful ρ17 3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ32 ζ3 2ζ98+ζ92 2ζ94+ζ9 2ζ97+ζ94 ζ97+2ζ9 ζ95+2ζ92 ζ98+2ζ95 ζ97 ζ9 ζ94 ζ95 ζ98 ζ92 complex faithful ρ18 3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ3 ζ32 ζ97+2ζ9 ζ98+2ζ95 ζ95+2ζ92 2ζ98+ζ92 2ζ97+ζ94 2ζ94+ζ9 ζ92 ζ98 ζ95 ζ94 ζ9 ζ97 complex faithful ρ19 3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ32 ζ3 ζ95+2ζ92 ζ97+2ζ9 2ζ94+ζ9 2ζ97+ζ94 ζ98+2ζ95 2ζ98+ζ92 ζ94 ζ97 ζ9 ζ98 ζ92 ζ95 complex faithful ρ20 3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ3 ζ32 2ζ97+ζ94 2ζ98+ζ92 ζ98+2ζ95 ζ95+2ζ92 2ζ94+ζ9 ζ97+2ζ9 ζ95 ζ92 ζ98 ζ9 ζ97 ζ94 complex faithful ρ21 3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ3 ζ32 2ζ94+ζ9 ζ95+2ζ92 2ζ98+ζ92 ζ98+2ζ95 ζ97+2ζ9 2ζ97+ζ94 ζ98 ζ95 ζ92 ζ97 ζ94 ζ9 complex faithful ρ22 6 0 6 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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)(11 17 23)(13 19 25)(15 21 27)
(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 25 16)(2 23 14)(3 21 12)(4 19 10)(5 17 26)(6 15 24)(7 13 22)(8 11 20)(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)

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

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

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

G:=TransitiveGroup(27,40);

On 27 points - transitive group 27T49
Generators in S27
(1 19 22)(2 11 14)(3 21 24)(4 13 16)(5 23 26)(6 15 18)(7 25 10)(8 17 20)(9 27 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)
(1 19 10)(2 14 23)(4 13 22)(5 26 17)(7 25 16)(8 20 11)(12 24 18)(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)| (1,19,22)(2,11,14)(3,21,24)(4,13,16)(5,23,26)(6,15,18)(7,25,10)(8,17,20)(9,27,12), (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), (1,19,10)(2,14,23)(4,13,22)(5,26,17)(7,25,16)(8,20,11)(12,24,18)(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( (1,19,22)(2,11,14)(3,21,24)(4,13,16)(5,23,26)(6,15,18)(7,25,10)(8,17,20)(9,27,12), (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), (1,19,10)(2,14,23)(4,13,22)(5,26,17)(7,25,16)(8,20,11)(12,24,18)(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([[(1,19,22),(2,11,14),(3,21,24),(4,13,16),(5,23,26),(6,15,18),(7,25,10),(8,17,20),(9,27,12)], [(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)], [(1,19,10),(2,14,23),(4,13,22),(5,26,17),(7,25,16),(8,20,11),(12,24,18),(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(27,49);

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

 0 1 0 0 0 1 1 0 0
,
 7 0 0 0 7 0 0 0 7
,
 0 11 0 0 0 7 1 0 0
,
 13 10 13 13 15 15 10 10 15
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

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