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## G = He3.4C6order 162 = 2·34

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

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

 Derived series C1 — C3 — He3 — He3.4C6
 Chief series C1 — C3 — C32 — He3 — C9○He3 — He3.4C6
 Lower central He3 — He3.4C6
 Upper central C1 — C9

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 >

Character table of He3.4C6

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L 9M 9N 18A 18B 18C 18D 18E 18F size 1 9 1 1 6 6 6 6 9 9 1 1 1 1 1 1 6 6 6 6 6 6 6 6 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 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 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 1 1 1 1 1 1 -1 -1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 linear of order 6 ρ4 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ5 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ6 1 -1 1 1 1 1 1 1 -1 -1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 linear of order 6 ρ7 2 0 2 2 -1 -1 2 -1 0 0 2 2 2 2 2 2 -1 -1 -1 -1 2 2 -1 -1 0 0 0 0 0 0 orthogonal lifted from S3 ρ8 2 0 2 2 2 -1 -1 -1 0 0 2 2 2 2 2 2 -1 -1 2 -1 -1 -1 -1 2 0 0 0 0 0 0 orthogonal lifted from S3 ρ9 2 0 2 2 -1 -1 -1 2 0 0 2 2 2 2 2 2 -1 2 -1 -1 -1 -1 2 -1 0 0 0 0 0 0 orthogonal lifted from S3 ρ10 2 0 2 2 -1 2 -1 -1 0 0 2 2 2 2 2 2 2 -1 -1 2 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from S3 ρ11 2 0 2 2 -1 -1 2 -1 0 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 -1-√-3 -1+√-3 ζ6 ζ65 ζ65 ζ65 -1+√-3 -1-√-3 ζ6 ζ6 0 0 0 0 0 0 complex lifted from C3×S3 ρ12 2 0 2 2 -1 -1 -1 2 0 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 -1+√-3 -1-√-3 ζ65 -1-√-3 ζ6 ζ6 ζ6 ζ65 -1+√-3 ζ65 0 0 0 0 0 0 complex lifted from C3×S3 ρ13 2 0 2 2 -1 -1 -1 2 0 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 -1-√-3 -1+√-3 ζ6 -1+√-3 ζ65 ζ65 ζ65 ζ6 -1-√-3 ζ6 0 0 0 0 0 0 complex lifted from C3×S3 ρ14 2 0 2 2 -1 2 -1 -1 0 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 -1-√-3 -1+√-3 -1-√-3 ζ65 ζ65 -1+√-3 ζ65 ζ6 ζ6 ζ6 0 0 0 0 0 0 complex lifted from C3×S3 ρ15 2 0 2 2 2 -1 -1 -1 0 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 -1+√-3 -1-√-3 ζ65 ζ6 -1-√-3 ζ6 ζ6 ζ65 ζ65 -1+√-3 0 0 0 0 0 0 complex lifted from C3×S3 ρ16 2 0 2 2 -1 -1 2 -1 0 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 -1+√-3 -1-√-3 ζ65 ζ6 ζ6 ζ6 -1-√-3 -1+√-3 ζ65 ζ65 0 0 0 0 0 0 complex lifted from C3×S3 ρ17 2 0 2 2 -1 2 -1 -1 0 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 -1+√-3 -1-√-3 -1+√-3 ζ6 ζ6 -1-√-3 ζ6 ζ65 ζ65 ζ65 0 0 0 0 0 0 complex lifted from C3×S3 ρ18 2 0 2 2 2 -1 -1 -1 0 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 -1-√-3 -1+√-3 ζ6 ζ65 -1+√-3 ζ65 ζ65 ζ6 ζ6 -1-√-3 0 0 0 0 0 0 complex lifted from C3×S3 ρ19 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ6 ζ65 3ζ92 3ζ94 3ζ95 3ζ97 3ζ9 3ζ98 0 0 0 0 0 0 0 0 -ζ94 -ζ97 -ζ9 -ζ98 -ζ92 -ζ95 complex faithful ρ20 3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ32 ζ3 3ζ95 3ζ9 3ζ98 3ζ94 3ζ97 3ζ92 0 0 0 0 0 0 0 0 ζ9 ζ94 ζ97 ζ92 ζ95 ζ98 complex faithful ρ21 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ65 ζ6 3ζ97 3ζ95 3ζ94 3ζ92 3ζ98 3ζ9 0 0 0 0 0 0 0 0 -ζ95 -ζ92 -ζ98 -ζ9 -ζ97 -ζ94 complex faithful ρ22 3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ3 ζ32 3ζ97 3ζ95 3ζ94 3ζ92 3ζ98 3ζ9 0 0 0 0 0 0 0 0 ζ95 ζ92 ζ98 ζ9 ζ97 ζ94 complex faithful ρ23 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ6 ζ65 3ζ95 3ζ9 3ζ98 3ζ94 3ζ97 3ζ92 0 0 0 0 0 0 0 0 -ζ9 -ζ94 -ζ97 -ζ92 -ζ95 -ζ98 complex faithful ρ24 3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ32 ζ3 3ζ98 3ζ97 3ζ92 3ζ9 3ζ94 3ζ95 0 0 0 0 0 0 0 0 ζ97 ζ9 ζ94 ζ95 ζ98 ζ92 complex faithful ρ25 3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ3 ζ32 3ζ9 3ζ92 3ζ97 3ζ98 3ζ95 3ζ94 0 0 0 0 0 0 0 0 ζ92 ζ98 ζ95 ζ94 ζ9 ζ97 complex faithful ρ26 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ6 ζ65 3ζ98 3ζ97 3ζ92 3ζ9 3ζ94 3ζ95 0 0 0 0 0 0 0 0 -ζ97 -ζ9 -ζ94 -ζ95 -ζ98 -ζ92 complex faithful ρ27 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ65 ζ6 3ζ9 3ζ92 3ζ97 3ζ98 3ζ95 3ζ94 0 0 0 0 0 0 0 0 -ζ92 -ζ98 -ζ95 -ζ94 -ζ9 -ζ97 complex faithful ρ28 3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ3 ζ32 3ζ94 3ζ98 3ζ9 3ζ95 3ζ92 3ζ97 0 0 0 0 0 0 0 0 ζ98 ζ95 ζ92 ζ97 ζ94 ζ9 complex faithful ρ29 3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ32 ζ3 3ζ92 3ζ94 3ζ95 3ζ97 3ζ9 3ζ98 0 0 0 0 0 0 0 0 ζ94 ζ97 ζ9 ζ98 ζ92 ζ95 complex faithful ρ30 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ65 ζ6 3ζ94 3ζ98 3ζ9 3ζ95 3ζ92 3ζ97 0 0 0 0 0 0 0 0 -ζ98 -ζ95 -ζ92 -ζ97 -ζ94 -ζ9 complex faithful

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

 1 0 0 0 11 0 0 0 7
,
 11 0 0 0 11 0 0 0 11
,
 0 1 0 0 0 1 1 0 0
,
 6 0 0 0 0 6 0 6 0
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

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