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G = He3.3S3order 162 = 2·34

3rd non-split extension by He3 of S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He3.3S3, 3- 1+22S3, (C3×C9)⋊5S3, He3.C31C2, C32.2(C3⋊S3), C3.3(He3⋊C2), SmallGroup(162,20)

Series: Derived Chief Lower central Upper central

C1C32He3.C3 — He3.3S3
C1C3C32C3×C9He3.C3 — He3.3S3
He3.C3 — He3.3S3
C1

Generators and relations for He3.3S3
 G = < a,b,c,d,e | a3=b3=c3=e2=1, d3=ebe=b-1, ab=ba, cac-1=eae=ab-1, ad=da, bc=cb, bd=db, dcd-1=a-1bc, ece=c-1, ede=bd2 >

27C2
3C3
9C3
9S3
27S3
27C6
3C9
3C9
3C32
3C9
3D9
3D9
3C3⋊S3
3D9
9C3×S3
3C9⋊C6
3C3×D9
3C32⋊C6
3C9⋊C6

Character table of He3.3S3

 class 123A3B3C3D6A6B9A9B9C9D9E
 size 1272331827276661818
ρ11111111111111    trivial
ρ21-11111-1-111111    linear of order 2
ρ320222-100-1-1-12-1    orthogonal lifted from S3
ρ420222-100222-1-1    orthogonal lifted from S3
ρ520222200-1-1-1-1-1    orthogonal lifted from S3
ρ620222-100-1-1-1-12    orthogonal lifted from S3
ρ7313-3+3-3/2-3-3-3/20ζ3ζ3200000    complex lifted from He3⋊C2
ρ8313-3-3-3/2-3+3-3/20ζ32ζ300000    complex lifted from He3⋊C2
ρ93-13-3+3-3/2-3-3-3/20ζ65ζ600000    complex lifted from He3⋊C2
ρ103-13-3-3-3/2-3+3-3/20ζ6ζ6500000    complex lifted from He3⋊C2
ρ1160-3000009594929ζ989794+2ζ92ζ989492+2ζ900    orthogonal faithful
ρ1260-300000ζ989492+2ζ99594929ζ989794+2ζ9200    orthogonal faithful
ρ1360-300000ζ989794+2ζ92ζ989492+2ζ9959492900    orthogonal faithful

Permutation representations of He3.3S3
On 27 points - transitive group 27T42
Generators in S27
(1 4 7)(2 5 8)(3 6 9)(19 25 22)(20 26 23)(21 27 24)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)
(1 22 13)(2 26 17)(3 21 12)(4 25 16)(5 20 11)(6 24 15)(7 19 10)(8 23 14)(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)
(2 9)(3 8)(4 7)(5 6)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(17 27)(18 26)

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

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

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

G:=TransitiveGroup(27,42);

On 27 points - transitive group 27T68
Generators in S27
(1 11 24)(2 12 25)(3 13 26)(4 14 27)(5 15 19)(6 16 20)(7 17 21)(8 18 22)(9 10 23)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)
(2 25 15)(3 13 23)(5 19 18)(6 16 26)(8 22 12)(9 10 20)(11 17 14)(21 24 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)
(2 9)(3 8)(4 7)(5 6)(10 15)(11 14)(12 13)(16 18)(19 26)(20 25)(21 24)(22 23)

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

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

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

G:=TransitiveGroup(27,68);

He3.3S3 is a maximal subgroup of
He3.D6  He3.C32C6  He3.(C3⋊S3)  C3≀C3⋊S3
He3.3S3 is a maximal quotient of
He3.3Dic3  (C3×He3).S3  C33.(C3⋊S3)  C32⋊C96S3  C3.(He3⋊S3)  C32⋊C9.10S3  (C3×C9)⋊5D9  He32D9  3- 1+2⋊D9  He3.(C3⋊S3)

Matrix representation of He3.3S3 in GL6(𝔽19)

001000
000100
000010
000001
100000
010000
,
010000
18180000
000100
00181800
000001
00001818
,
100000
010000
00181800
001000
000001
00001818
,
1815181543
4343161
4318151815
1614343
1815431815
4316143
,
100000
18180000
00181800
000100
000001
000010

G:=sub<GL(6,GF(19))| [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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,1,18],[18,4,4,16,18,4,15,3,3,1,15,3,18,4,18,4,4,16,15,3,15,3,3,1,4,16,18,4,18,4,3,1,15,3,15,3],[1,18,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,18,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

He3.3S3 in GAP, Magma, Sage, TeX

{\rm He}_3._3S_3
% in TeX

G:=Group("He3.3S3");
// GroupNames label

G:=SmallGroup(162,20);
// by ID

G=gap.SmallGroup(162,20);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,41,1802,187,147,2523,728,2704]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^3=e*b*e=b^-1,a*b=b*a,c*a*c^-1=e*a*e=a*b^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*b*c,e*c*e=c^-1,e*d*e=b*d^2>;
// generators/relations

Export

Subgroup lattice of He3.3S3 in TeX
Character table of He3.3S3 in TeX

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