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G = He3⋊S3order 162 = 2·34

3rd semidirect product of He3 and S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He33S3, (C3×C9)⋊6S3, He3⋊C33C2, C32.3(C3⋊S3), C3.4(He3⋊C2), SmallGroup(162,21)

Series: Derived Chief Lower central Upper central

C1C32He3⋊C3 — He3⋊S3
C1C3C32He3He3⋊C3 — He3⋊S3
He3⋊C3 — He3⋊S3
C1

Generators and relations for He3⋊S3
 G = < a,b,c,d,e | a3=b3=c3=d3=e2=1, dad-1=ab=ba, cac-1=ab-1, ae=ea, bc=cb, bd=db, ebe=b-1, dcd-1=a-1b-1c, ece=ab-1c-1, ede=d-1 >

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

Character table of He3⋊S3

 class 123A3B3C3D3E3F6A6B9A9B9C
 size 1272331818182727666
ρ11111111111111    trivial
ρ21-1111111-1-1111    linear of order 2
ρ320222-1-1-100222    orthogonal lifted from S3
ρ420222-1-1200-1-1-1    orthogonal lifted from S3
ρ5202222-1-100-1-1-1    orthogonal lifted from S3
ρ620222-12-100-1-1-1    orthogonal lifted from S3
ρ7313-3-3-3/2-3+3-3/2000ζ3ζ32000    complex lifted from He3⋊C2
ρ83-13-3+3-3/2-3-3-3/2000ζ6ζ65000    complex lifted from He3⋊C2
ρ93-13-3-3-3/2-3+3-3/2000ζ65ζ6000    complex lifted from He3⋊C2
ρ10313-3+3-3/2-3-3-3/2000ζ32ζ3000    complex lifted from He3⋊C2
ρ1160-30000000989492998+2ζ979492ζ95+2ζ94929    orthogonal faithful
ρ1260-30000000ζ95+2ζ94929989492998+2ζ979492    orthogonal faithful
ρ1360-3000000098+2ζ979492ζ95+2ζ949299894929    orthogonal faithful

Permutation representations of He3⋊S3
On 27 points - transitive group 27T44
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 5 4)(2 3 6)(7 8 9)(10 12 11)(13 14 15)(16 18 17)(19 20 21)(22 24 23)(25 26 27)
(1 16 19)(2 24 26)(3 23 27)(4 17 21)(5 18 20)(6 22 25)(7 11 15)(8 10 13)(9 12 14)
(1 27 10)(2 14 18)(3 15 17)(4 26 11)(5 25 12)(6 13 16)(7 20 24)(8 21 23)(9 19 22)
(1 9)(2 3)(4 7)(5 8)(10 19)(11 20)(12 21)(13 16)(14 17)(15 18)(22 27)(23 25)(24 26)

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

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

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

G:=TransitiveGroup(27,44);

On 27 points - transitive group 27T66
Generators in S27
(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)
(1 12 15)(2 10 13)(3 11 14)(4 26 8)(5 27 9)(6 25 7)(16 21 23)(17 19 24)(18 20 22)
(1 3 13)(2 12 11)(5 27 9)(6 7 25)(10 15 14)(16 19 18)(17 22 23)(20 21 24)
(1 20 9)(2 16 6)(3 24 26)(4 14 19)(5 12 22)(7 13 23)(8 11 17)(10 21 25)(15 18 27)
(4 17)(5 18)(6 16)(7 21)(8 19)(9 20)(10 13)(11 14)(12 15)(22 27)(23 25)(24 26)

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

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

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

G:=TransitiveGroup(27,66);

He3⋊S3 is a maximal subgroup of
He3.2D6  C92⋊S3  C9⋊C9⋊S3  He3⋊(C3×S3)  C3⋊(He3⋊S3)  C3≀C3⋊S3
He3⋊S3 is a maximal quotient of
He3⋊Dic3  (C3×He3)⋊S3  C32⋊C96S3  C3.(He3⋊S3)  (C3×C9)⋊6D9  He32D9  C922S3  C3⋊(He3⋊S3)

Matrix representation of He3⋊S3 in GL6(𝔽19)

0018100
77171800
0012010
0012001
008000
108000
,
0180000
1180000
70181800
0121000
110001818
080010
,
757575
272727
1016112610
711185112
121139112
40112187
,
531421712
1621214517
81478413
15014131118
614104108
7317981
,
010000
100000
001000
77181800
000010
1111001818

G:=sub<GL(6,GF(19))| [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],[0,1,7,0,11,0,18,18,0,12,0,8,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0],[7,2,10,7,12,4,5,7,16,11,1,0,7,2,1,18,13,11,5,7,12,5,9,2,7,2,6,1,11,18,5,7,10,12,2,7],[5,16,8,15,6,7,3,2,14,0,14,3,14,12,7,14,10,17,2,14,8,13,4,9,17,5,4,11,10,8,12,17,13,18,8,1],[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\rtimes S_3
% in TeX

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

G:=SmallGroup(162,21);
// by ID

G=gap.SmallGroup(162,21);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,41,182,187,728,433,2704]);
// Polycyclic

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

Export

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

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