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## G = He3.2S3order 162 = 2·34

### 2nd non-split extension by He3 of S3 acting faithfully

Aliases: He3.2S3, C9⋊S33C3, (C3×C9)⋊3C6, He3⋊C32C2, C32.8(C3×S3), C3.4(C32⋊C6), SmallGroup(162,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — He3.2S3
 Chief series C1 — C3 — C32 — C3×C9 — He3⋊C3 — He3.2S3
 Lower central C3×C9 — He3.2S3
 Upper central C1

Generators and relations for He3.2S3
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, dcd-1=a-1bc, ce=ec, ede=b-1d2 >

27C2
3C3
9C3
18C3
9S3
27C6
27S3
3C9
3C32
6C32
9D9
2He3

Character table of He3.2S3

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 9A 9B 9C size 1 27 2 6 9 9 18 18 27 27 6 6 6 ρ1 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 linear of order 2 ρ3 1 -1 1 1 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 1 1 1 linear of order 6 ρ4 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 1 linear of order 3 ρ5 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 1 linear of order 3 ρ6 1 -1 1 1 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 1 1 1 linear of order 6 ρ7 2 0 2 2 2 2 -1 -1 0 0 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 2 2 -1-√-3 -1+√-3 ζ65 ζ6 0 0 -1 -1 -1 complex lifted from C3×S3 ρ9 2 0 2 2 -1+√-3 -1-√-3 ζ6 ζ65 0 0 -1 -1 -1 complex lifted from C3×S3 ρ10 6 0 6 -3 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ11 6 0 -3 0 0 0 0 0 0 0 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 orthogonal faithful ρ12 6 0 -3 0 0 0 0 0 0 0 -ζ98+2ζ97+ζ94+ζ92 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 orthogonal faithful ρ13 6 0 -3 0 0 0 0 0 0 0 2ζ98-ζ94+ζ92+ζ9 ζ95+2ζ94-ζ92+ζ9 -ζ98+2ζ97+ζ94+ζ92 orthogonal faithful

Permutation representations of He3.2S3
On 27 points - transitive group 27T38
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 25 13)(2 20 14)(3 24 15)(4 19 16)(5 23 17)(6 27 18)(7 22 10)(8 26 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 22)(20 21)(23 27)(24 26)

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

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

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

G:=TransitiveGroup(27,38);

On 27 points - transitive group 27T64
Generators in S27
(1 14 25)(2 15 26)(3 16 27)(4 17 19)(5 18 20)(6 10 21)(7 11 22)(8 12 23)(9 13 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)
(2 26 12)(3 16 21)(5 20 15)(6 10 24)(8 23 18)(9 13 27)(11 14 17)(19 25 22)
(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 20)(11 19)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)

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

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

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

G:=TransitiveGroup(27,64);

He3.2S3 is a maximal subgroup of
He3.2D6  C92⋊C6  C922C6  He3.(C3×S3)  He3⋊C33S3  C3≀C3.S3
He3.2S3 is a maximal quotient of
He3.2Dic3  C32⋊C9.S3  C3.3C3≀S3  C33.(C3×S3)  C9⋊S33C9  He3⋊D9  C92⋊C6  C922C6  He3⋊C33S3

Matrix representation of He3.2S3 in GL6(𝔽19)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 14 9 18 18 0 0 5 2 0 0 18 18 0 0 0 0 1 0
,
 18 1 0 0 0 0 18 0 0 0 0 0 13 8 0 1 0 0 8 17 18 18 0 0 15 1 0 0 0 1 1 3 0 0 18 18
,
 0 0 18 1 0 0 14 9 17 18 0 0 15 1 2 8 1 0 15 1 2 8 0 1 9 12 16 1 0 0 10 12 16 1 0 0
,
 5 12 0 0 0 0 7 17 0 0 0 0 4 1 17 12 0 0 1 14 7 5 0 0 18 8 0 0 14 2 9 12 0 0 17 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 11 8 1 0 0 0 6 17 18 18 0 0 18 1 0 0 1 0 4 3 0 0 18 18

G:=sub<GL(6,GF(19))| [1,0,0,14,5,0,0,1,0,9,2,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[18,18,13,8,15,1,1,0,8,17,1,3,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[0,14,15,15,9,10,0,9,1,1,12,12,18,17,2,2,16,16,1,18,8,8,1,1,0,0,1,0,0,0,0,0,0,1,0,0],[5,7,4,1,18,9,12,17,1,14,8,12,0,0,17,7,0,0,0,0,12,5,0,0,0,0,0,0,14,17,0,0,0,0,2,12],[0,1,11,6,18,4,1,0,8,17,1,3,0,0,1,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,18] >;

He3.2S3 in GAP, Magma, Sage, TeX

{\rm He}_3._2S_3
% in TeX

G:=Group("He3.2S3");
// GroupNames label

G:=SmallGroup(162,15);
// by ID

G=gap.SmallGroup(162,15);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,992,187,282,723,728,2704]);
// 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,d*c*d^-1=a^-1*b*c,c*e=e*c,e*d*e=b^-1*d^2>;
// generators/relations

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