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## G = He3.4S3order 162 = 2·34

### The non-split extension by He3 of S3 acting via S3/C3=C2

Aliases: He3.4S3, 3- 1+24S3, C9.(C3×S3), C9⋊S35C3, (C3×C9)⋊6C6, (C3×C9)⋊7S3, C9○He31C2, C32.9(C3×S3), C32.6(C3⋊S3), C3.4(C3×C3⋊S3), SmallGroup(162,43)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — He3.4S3
 Chief series C1 — C3 — C32 — C3×C9 — C9○He3 — He3.4S3
 Lower central C3×C9 — He3.4S3
 Upper central C1

Generators and relations for He3.4S3
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, cd=dc, ce=ec, ede=b-1d2 >

Character table of He3.4S3

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K size 1 27 2 3 3 6 6 6 27 27 2 2 2 6 6 6 6 6 6 6 6 ρ1 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 linear of order 2 ρ3 1 1 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 1 1 1 ζ3 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 linear of order 3 ρ4 1 -1 1 ζ32 ζ3 1 ζ32 ζ3 ζ6 ζ65 1 1 1 ζ32 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 linear of order 6 ρ5 1 1 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 1 1 1 ζ32 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 linear of order 3 ρ6 1 -1 1 ζ3 ζ32 1 ζ3 ζ32 ζ65 ζ6 1 1 1 ζ3 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 linear of order 6 ρ7 2 0 2 2 2 -1 -1 -1 0 0 -1 -1 -1 -1 -1 -1 -1 2 2 2 -1 orthogonal lifted from S3 ρ8 2 0 2 2 2 -1 -1 -1 0 0 2 2 2 2 -1 2 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ9 2 0 2 2 2 -1 -1 -1 0 0 -1 -1 -1 -1 2 -1 2 -1 -1 -1 2 orthogonal lifted from S3 ρ10 2 0 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 0 2 -1-√-3 -1+√-3 2 -1-√-3 -1+√-3 0 0 -1 -1 -1 ζ6 -1 ζ65 ζ6 ζ6 -1 ζ65 ζ65 complex lifted from C3×S3 ρ12 2 0 2 -1-√-3 -1+√-3 -1 ζ6 ζ65 0 0 -1 -1 -1 ζ6 2 ζ65 -1-√-3 ζ6 -1 ζ65 -1+√-3 complex lifted from C3×S3 ρ13 2 0 2 -1+√-3 -1-√-3 -1 ζ65 ζ6 0 0 -1 -1 -1 ζ65 -1 ζ6 ζ65 -1+√-3 2 -1-√-3 ζ6 complex lifted from C3×S3 ρ14 2 0 2 -1-√-3 -1+√-3 -1 ζ6 ζ65 0 0 2 2 2 -1-√-3 -1 -1+√-3 ζ6 ζ6 -1 ζ65 ζ65 complex lifted from C3×S3 ρ15 2 0 2 -1+√-3 -1-√-3 2 -1+√-3 -1-√-3 0 0 -1 -1 -1 ζ65 -1 ζ6 ζ65 ζ65 -1 ζ6 ζ6 complex lifted from C3×S3 ρ16 2 0 2 -1-√-3 -1+√-3 -1 ζ6 ζ65 0 0 -1 -1 -1 ζ6 -1 ζ65 ζ6 -1-√-3 2 -1+√-3 ζ65 complex lifted from C3×S3 ρ17 2 0 2 -1+√-3 -1-√-3 -1 ζ65 ζ6 0 0 -1 -1 -1 ζ65 2 ζ6 -1+√-3 ζ65 -1 ζ6 -1-√-3 complex lifted from C3×S3 ρ18 2 0 2 -1+√-3 -1-√-3 -1 ζ65 ζ6 0 0 2 2 2 -1+√-3 -1 -1-√-3 ζ65 ζ65 -1 ζ6 ζ6 complex lifted from C3×S3 ρ19 6 0 -3 0 0 0 0 0 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 0 0 0 0 0 0 0 0 orthogonal faithful ρ20 6 0 -3 0 0 0 0 0 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 0 0 0 0 0 0 0 0 orthogonal faithful ρ21 6 0 -3 0 0 0 0 0 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of He3.4S3
On 27 points - transitive group 27T54
Generators in S27
(1 14 26)(2 15 27)(3 16 19)(4 17 20)(5 18 21)(6 10 22)(7 11 23)(8 12 24)(9 13 25)
(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)
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)
(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 21)(11 20)(12 19)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)

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

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

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

G:=TransitiveGroup(27,54);

On 27 points - transitive group 27T56
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 22 13)(2 23 14)(3 24 15)(4 25 16)(5 26 17)(6 27 18)(7 19 10)(8 20 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 25)(20 24)(21 23)(26 27)

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

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

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

G:=TransitiveGroup(27,56);

He3.4S3 is a maximal subgroup of
He3.6D6  He3.D9  He3.2D9  He3.3D9  He3.4D9  He3.5D9  C3≀C3.S3  C3≀C3⋊S3  3- 1+4⋊C2  C9○He33S3
He3.4S3 is a maximal quotient of
He3.4Dic3  C923S3  C923C6  He33D9  C929C6  C9⋊He32C2  (C32×C9)⋊S3  (C32×C9)⋊C6  C926S3  C9210C6  C924C6  C925S3  C925C6  C9211C6  C9○He33S3

Matrix representation of He3.4S3 in GL6(𝔽19)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 18 18 0 0 0 0 0 0 18 18 0 0 0 0 1 0
,
 0 1 0 0 0 0 18 18 0 0 0 0 0 0 0 1 0 0 0 0 18 18 0 0 0 0 0 0 0 1 0 0 0 0 18 18
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0
,
 14 2 0 0 0 0 17 12 0 0 0 0 0 0 14 2 0 0 0 0 17 12 0 0 0 0 0 0 14 2 0 0 0 0 17 12
,
 1 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 18 18

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[14,17,0,0,0,0,2,12,0,0,0,0,0,0,14,17,0,0,0,0,2,12,0,0,0,0,0,0,14,17,0,0,0,0,2,12],[1,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,18] >;

He3.4S3 in GAP, Magma, Sage, TeX

{\rm He}_3._4S_3
% in TeX

G:=Group("He3.4S3");
// GroupNames label

G:=SmallGroup(162,43);
// by ID

G=gap.SmallGroup(162,43);
# by ID

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

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