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G = He3⋊4S3  order 162 = 2·34

1st semidirect product of He3 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: He3⋊4S3, C33⋊4C6, C33⋊4S3, C3⋊(C32⋊C6), (C3×He3)⋊3C2, C32⋊3(C3×S3), C32⋊1(C3⋊S3), C33⋊C2⋊2C3, C3.2(C3×C3⋊S3), SmallGroup(162,40)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — He3⋊4S3
Chief series
C1 — C3 — C32 — C33 — C3×He3 — He3⋊4S3
Lower central
C33 — He3⋊4S3
Upper central
C1

Generators and relations for He3⋊4S3
 G = < a,b,c,d,e | a3=b3=c3=d3=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 364 in 67 conjugacy classes, 18 normal (9 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3⋊S3, He3, He3, C33, C33, C32⋊C6, C3×C3⋊S3, C33⋊C2, C3×He3, He3⋊4S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, He3⋊4S3

Character table of He3⋊4S3

 class 123A3B3C3D3E3F3G3H3I3J3K3L3M3N3O3P3Q6A6B
 size 127222233666666666662727
ρ1111111111111111111111    trivial
ρ21-111111111111111111-1-1    linear of order 2
ρ3111111ζ32ζ3ζ32ζ321ζ32ζ321ζ3ζ3ζ3ζ31ζ32ζ3    linear of order 3
ρ41-11111ζ3ζ32ζ3ζ31ζ3ζ31ζ32ζ32ζ32ζ321ζ65ζ6    linear of order 6
ρ51-11111ζ32ζ3ζ32ζ321ζ32ζ321ζ3ζ3ζ3ζ31ζ6ζ65    linear of order 6
ρ6111111ζ3ζ32ζ3ζ31ζ3ζ31ζ32ζ32ζ32ζ321ζ3ζ32    linear of order 3
ρ720-1-12-122-1-1-1-122-1-1-12-100    orthogonal lifted from S3
ρ820222222-12-1-1-1-12-1-1-1-100    orthogonal lifted from S3
ρ920-1-12-122-1-122-1-1-1-12-1-100    orthogonal lifted from S3
ρ1020-1-12-1222-1-1-1-1-1-12-1-1200    orthogonal lifted from S3
ρ1120-1-12-1-1-√-3-1+√-3ζ6ζ6-1ζ6-1-√-32ζ65ζ65ζ65-1+√-3-100    complex lifted from C3×S3
ρ1220-1-12-1-1+√-3-1-√-3ζ65ζ65-1ζ65-1+√-32ζ6ζ6ζ6-1-√-3-100    complex lifted from C3×S3
ρ13202222-1-√-3-1+√-3ζ6-1-√-3-1ζ6ζ6-1-1+√-3ζ65ζ65ζ65-100    complex lifted from C3×S3
ρ1420-1-12-1-1+√-3-1-√-3-1+√-3ζ65-1ζ65ζ65-1ζ6-1-√-3ζ6ζ6200    complex lifted from C3×S3
ρ1520-1-12-1-1-√-3-1+√-3ζ6ζ62-1-√-3ζ6-1ζ65ζ65-1+√-3ζ65-100    complex lifted from C3×S3
ρ1620-1-12-1-1-√-3-1+√-3-1-√-3ζ6-1ζ6ζ6-1ζ65-1+√-3ζ65ζ65200    complex lifted from C3×S3
ρ17202222-1+√-3-1-√-3ζ65-1+√-3-1ζ65ζ65-1-1-√-3ζ6ζ6ζ6-100    complex lifted from C3×S3
ρ1820-1-12-1-1+√-3-1-√-3ζ65ζ652-1+√-3ζ65-1ζ6ζ6-1-√-3ζ6-100    complex lifted from C3×S3
ρ19606-3-3-3000000000000000    orthogonal lifted from C32⋊C6
ρ2060-36-3-3000000000000000    orthogonal lifted from C32⋊C6
ρ2160-3-3-36000000000000000    orthogonal lifted from C32⋊C6

Permutation representations of He3⋊4S3
►On 27 points - transitive group 27T61
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 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)

(2 27 9)(3 7 25)(4 19 23)(5 24 20)(10 18 15)(11 13 16)

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

(2 3)(4 13)(5 15)(6 14)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(16 19)(17 21)(18 20)
 
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), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (2,27,9)(3,7,25)(4,19,23)(5,24,20)(10,18,15)(11,13,16), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (2,3)(4,13)(5,15)(6,14)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(16,19)(17,21)(18,20)>;
 
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), (1,26,8)(2,27,9)(3,25,7)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (2,27,9)(3,7,25)(4,19,23)(5,24,20)(10,18,15)(11,13,16), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (2,3)(4,13)(5,15)(6,14)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(16,19)(17,21)(18,20) );
 
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)], [(1,26,8),(2,27,9),(3,25,7),(4,19,23),(5,20,24),(6,21,22),(10,18,15),(11,16,13),(12,17,14)], [(2,27,9),(3,7,25),(4,19,23),(5,24,20),(10,18,15),(11,13,16)], [(1,14,6),(2,15,4),(3,13,5),(7,16,24),(8,17,22),(9,18,23),(10,19,27),(11,20,25),(12,21,26)], [(2,3),(4,13),(5,15),(6,14),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(16,19),(17,21),(18,20)]])
 
G:=TransitiveGroup(27,61);
 

►On 27 points - transitive group 27T71
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)

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

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

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

(2 3)(4 7)(5 9)(6 8)(11 12)(13 16)(14 18)(15 17)(20 21)(22 25)(23 27)(24 26)
 
G:=sub<Sym(27)| (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26), (1,19,10)(2,20,12)(3,21,11)(4,22,14)(5,23,13)(6,24,15)(7,25,18)(8,26,17)(9,27,16), (1,4,7)(2,5,8)(3,6,9)(10,14,18)(11,15,16)(12,13,17)(19,22,25)(20,23,26)(21,24,27), (2,3)(4,7)(5,9)(6,8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26)>;
 
G:=Group( (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26), (1,19,10)(2,20,12)(3,21,11)(4,22,14)(5,23,13)(6,24,15)(7,25,18)(8,26,17)(9,27,16), (1,4,7)(2,5,8)(3,6,9)(10,14,18)(11,15,16)(12,13,17)(19,22,25)(20,23,26)(21,24,27), (2,3)(4,7)(5,9)(6,8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)(23,27)(24,26) );
 
G=PermutationGroup([[(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27)], [(1,3,2),(4,6,5),(7,9,8),(10,11,12),(13,14,15),(16,17,18),(19,21,20),(22,24,23),(25,27,26)], [(1,19,10),(2,20,12),(3,21,11),(4,22,14),(5,23,13),(6,24,15),(7,25,18),(8,26,17),(9,27,16)], [(1,4,7),(2,5,8),(3,6,9),(10,14,18),(11,15,16),(12,13,17),(19,22,25),(20,23,26),(21,24,27)], [(2,3),(4,7),(5,9),(6,8),(11,12),(13,16),(14,18),(15,17),(20,21),(22,25),(23,27),(24,26)]])
 
G:=TransitiveGroup(27,71);
 

He3⋊4S3 is a maximal subgroup of
 S3×C32⋊C6  He3⋊5D6  He3⋊6D6  He3⋊D9  (C3×He3)⋊S3  (C3×He3).S3  C32⋊C9⋊6S3  He3⋊2D9  C34⋊S3  (C3×He3)⋊C6  He3⋊3D9  C34⋊4C6  C9⋊He3⋊2C2  (C32×C9)⋊C6  C34⋊5C6  C32⋊4D9⋊C3  He3⋊C3⋊3S3  C34⋊7S3  He3.(C3⋊S3)  C3⋊(He3⋊S3)  3+ 1+4⋊C2  C34⋊10C6  C9○He3⋊3S3
He3⋊4S3 is a maximal quotient of
 C33⋊4C12  C33⋊C18  C33⋊D9  He3⋊3D9  C34⋊3S3  C34⋊4C6  C9⋊He3⋊2C2  (C32×C9)⋊8S3  (C32×C9)⋊C6  C34⋊5S3  C34⋊5C6  He3.C3⋊S3  C32⋊4D9⋊C3  He3⋊C3⋊2S3  He3⋊C3⋊3S3  C3≀C3.S3  C34⋊10C6

Matrix representation of He3⋊4S3 ►in GL8(𝔽7)

66000000
10000000
00006100
00115600
00116010
00116001
00662000
00062000
,
10000000
01000000
00610000
00600000
00600100
00016600
00200001
00050066
,
40000000
04000000
00100000
00010000
00116600
00001000
00000001
00550066
,
66000000
10000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
10000000
00010000
00100000
00110010
00660066
00661000
00006600

G:=sub<GL(8,GF(7))| [6,1,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,1,1,6,0,0,0,0,1,1,1,6,6,0,0,6,5,6,6,2,2,0,0,1,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,6,0,2,0,0,0,1,0,0,1,0,5,0,0,0,0,0,6,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,1,6],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,1,0,0,5,0,0,0,1,1,0,0,5,0,0,0,0,6,1,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,1,6],[6,1,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,6,6,0,0,0,1,0,1,6,6,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,6,0,0,0,0,1,6,0,0,0,0,0,0,0,6,0,0] >;
 

He3⋊4S3 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_4S_3
 
% in TeX
 
G:=Group("He3:4S3");
 
// GroupNames label
 
G:=SmallGroup(162,40);
 
// by ID
 
G=gap.SmallGroup(162,40);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-3,-3,182,457,723,2704]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^2=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,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of He3⋊4S3 in TeX

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