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G = He3.S3order 162 = 2·34

1st non-split extension by He3 of S3 acting faithfully

metabelian, supersoluble, monomial

Aliases: He3.1S3, C9⋊S32C3, (C3×C9)⋊2C6, He3.C32C2, C32.7(C3×S3), C3.3(C32⋊C6), SmallGroup(162,13)

Series: Derived Chief Lower central Upper central

C1C3×C9 — He3.S3
C1C3C32C3×C9He3.C3 — He3.S3
C3×C9 — He3.S3
C1

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

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

Character table of He3.S3

 class 123A3B3C3D6A6B9A9B9C9D9E
 size 127269927276661818
ρ11111111111111    trivial
ρ21-11111-1-111111    linear of order 2
ρ31111ζ32ζ3ζ3ζ32111ζ3ζ32    linear of order 3
ρ41-111ζ3ζ32ζ6ζ65111ζ32ζ3    linear of order 6
ρ51-111ζ32ζ3ζ65ζ6111ζ3ζ32    linear of order 6
ρ61111ζ3ζ32ζ32ζ3111ζ32ζ3    linear of order 3
ρ720222200-1-1-1-1-1    orthogonal lifted from S3
ρ82022-1--3-1+-300-1-1-1ζ65ζ6    complex lifted from C3×S3
ρ92022-1+-3-1--300-1-1-1ζ6ζ65    complex lifted from C3×S3
ρ10606-3000000000    orthogonal lifted from C32⋊C6
ρ1160-3000009594929ζ989794+2ζ92ζ989492+2ζ900    orthogonal faithful
ρ1260-300000ζ989492+2ζ99594929ζ989794+2ζ9200    orthogonal faithful
ρ1360-300000ζ989794+2ζ92ζ989492+2ζ9959492900    orthogonal faithful

Permutation representations of He3.S3
On 27 points - transitive group 27T41
Generators in S27
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)
(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 25 10)(2 23 11)(3 21 12)(4 19 13)(5 26 14)(6 24 15)(7 22 16)(8 20 17)(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)(11 18)(12 17)(13 16)(14 15)(19 22)(20 21)(23 27)(24 26)

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

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

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

G:=TransitiveGroup(27,41);

On 27 points - transitive group 27T72
Generators in S27
(1 11 23)(2 12 24)(3 13 25)(4 14 26)(5 15 27)(6 16 19)(7 17 20)(8 18 21)(9 10 22)
(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 24 15)(3 13 22)(5 27 18)(6 16 25)(8 21 12)(9 10 19)(11 17 14)(20 23 26)
(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 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 27)(17 26)(18 25)

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

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

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

G:=TransitiveGroup(27,72);

He3.S3 is a maximal subgroup of   He3.D6  C9⋊S3⋊C32  C324D9⋊C3  C3≀C3.S3
He3.S3 is a maximal quotient of   He3.Dic3  C32⋊C9⋊C6  C32⋊C9.C6  C322D9.C3  (C3×C9)⋊C18  He3⋊D9  C324D9⋊C3

Matrix representation of He3.S3 in GL6(𝔽19)

100000
010000
000100
77181800
1111001818
000010
,
1810000
1800000
1200100
07181800
800001
011001818
,
0018100
77171800
0012010
0012001
008000
108000
,
1770000
1250000
805700
011121700
161700214
1800057
,
010000
100000
001000
77181800
000010
1111001818

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

He3.S3 in GAP, Magma, Sage, TeX

{\rm He}_3.S_3
% in TeX

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

G:=SmallGroup(162,13);
// by ID

G=gap.SmallGroup(162,13);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,1802,187,147,723,728,2704]);
// 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=a*b^-1,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*b*c,c*e=e*c,e*d*e=b*d^2>;
// generators/relations

Export

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

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