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G = He3.4S3order 162 = 2·34

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

metabelian, supersoluble, monomial

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

C1C3×C9 — He3.4S3
C1C3C32C3×C9C9○He3 — He3.4S3
C3×C9 — He3.4S3
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 >

27C2
3C3
3C3
6C3
9S3
27C6
27S3
2C32
2C9
2C9
2C9
3D9
3C3⋊S3
3D9
3D9
9C3×S3
2C3×C9
23- 1+2
23- 1+2
23- 1+2
3C9⋊C6
3C9⋊C6
3C3×D9
3C32⋊C6

Character table of He3.4S3

 class 123A3B3C3D3E3F6A6B9A9B9C9D9E9F9G9H9I9J9K
 size 127233666272722266666666
ρ1111111111111111111111    trivial
ρ21-1111111-1-111111111111    linear of order 2
ρ3111ζ3ζ321ζ3ζ32ζ3ζ32111ζ31ζ32ζ3ζ31ζ32ζ32    linear of order 3
ρ41-11ζ32ζ31ζ32ζ3ζ6ζ65111ζ321ζ3ζ32ζ321ζ3ζ3    linear of order 6
ρ5111ζ32ζ31ζ32ζ3ζ32ζ3111ζ321ζ3ζ32ζ321ζ3ζ3    linear of order 3
ρ61-11ζ3ζ321ζ3ζ32ζ65ζ6111ζ31ζ32ζ3ζ31ζ32ζ32    linear of order 6
ρ720222-1-1-100-1-1-1-1-1-1-1222-1    orthogonal lifted from S3
ρ820222-1-1-1002222-12-1-1-1-1-1    orthogonal lifted from S3
ρ920222-1-1-100-1-1-1-12-12-1-1-12    orthogonal lifted from S3
ρ102022222200-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ11202-1--3-1+-32-1--3-1+-300-1-1-1ζ6-1ζ65ζ6ζ6-1ζ65ζ65    complex lifted from C3×S3
ρ12202-1--3-1+-3-1ζ6ζ6500-1-1-1ζ62ζ65-1--3ζ6-1ζ65-1+-3    complex lifted from C3×S3
ρ13202-1+-3-1--3-1ζ65ζ600-1-1-1ζ65-1ζ6ζ65-1+-32-1--3ζ6    complex lifted from C3×S3
ρ14202-1--3-1+-3-1ζ6ζ6500222-1--3-1-1+-3ζ6ζ6-1ζ65ζ65    complex lifted from C3×S3
ρ15202-1+-3-1--32-1+-3-1--300-1-1-1ζ65-1ζ6ζ65ζ65-1ζ6ζ6    complex lifted from C3×S3
ρ16202-1--3-1+-3-1ζ6ζ6500-1-1-1ζ6-1ζ65ζ6-1--32-1+-3ζ65    complex lifted from C3×S3
ρ17202-1+-3-1--3-1ζ65ζ600-1-1-1ζ652ζ6-1+-3ζ65-1ζ6-1--3    complex lifted from C3×S3
ρ18202-1+-3-1--3-1ζ65ζ600222-1+-3-1-1--3ζ65ζ65-1ζ6ζ6    complex lifted from C3×S3
ρ1960-3000000098+3ζ997+3ζ9295+3ζ9400000000    orthogonal faithful
ρ2060-3000000095+3ζ9498+3ζ997+3ζ9200000000    orthogonal faithful
ρ2160-3000000097+3ζ9295+3ζ9498+3ζ900000000    orthogonal faithful

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

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

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

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

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)

100000
010000
000100
00181800
00001818
000010
,
010000
18180000
000100
00181800
000001
00001818
,
001000
000100
000010
000001
100000
010000
,
1420000
17120000
0014200
00171200
0000142
00001712
,
100000
18180000
001000
00181800
000010
00001818

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

Export

Subgroup lattice of He3.4S3 in TeX
Character table of He3.4S3 in TeX

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