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G = C34⋊7S3  order 486 = 2·35

7th semidirect product of C34 and S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C34⋊7S3, C3≀C3⋊1S3, He3⋊2(C3⋊S3), C3⋊(C33⋊S3), C33⋊6(C3⋊S3), (C3×He3)⋊15S3, C3.3(He3⋊5S3), 3- 1+2⋊1(C3⋊S3), (C3×3- 1+2)⋊10S3, C32.1(C33⋊C2), C32.27(He3⋊C2), (C3×C3≀C3)⋊2C2, SmallGroup(486,185)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C3≀C3 — C34⋊7S3
Chief series
C1 — C3 — C32 — C33 — C3≀C3 — C3×C3≀C3 — C34⋊7S3
Lower central
C3×C3≀C3 — C34⋊7S3
Upper central
C1

Generators and relations for C34⋊7S3
 G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, faf=a-1, ebe-1=bc=cb, bd=db, fbf=b-1, ece-1=cd=dc, fcf=cd-1, de=ed, fdf=d-1, fef=e-1 >

Subgroups: 1978 in 207 conjugacy classes, 35 normal (9 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, C33, C32⋊C6, C9⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, C3≀C3, C3×He3, C3×3- 1+2, C34, C33⋊S3, He3⋊4S3, C33.S3, C3×C33⋊C2, C3×C3≀C3, C34⋊7S3
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊C2, C33⋊S3, He3⋊5S3, C34⋊7S3

Character table of C34⋊7S3

 class 123A3B3C3D3E3F3G3H3I3J3K3L3M3N3O3P3Q3R3S3T6A6B9A9B9C9D9E9F
 size 181222233666666666661818188181181818181818
ρ1111111111111111111111111111111    trivial
ρ21-111111111111111111111-1-1111111    linear of order 2
ρ3202-1-1-122-1-1-1-1-1-1-1222-12-1-1002-1-1-12-1    orthogonal lifted from S3
ρ4202-1-1-122-1-1-1-1-1-1-1222-1-1-1200-1-122-1-1    orthogonal lifted from S3
ρ5202-1-1-122-12-1-12-1-1-1-1-122-1-100-1-1-12-12    orthogonal lifted from S3
ρ6202-1-1-122-1-1-1-1-1-1-1222-1-12-100-12-1-1-12    orthogonal lifted from S3
ρ7202-1-1-1222-1-1-1-122-1-1-1-1-12-1002-1-12-1-1    orthogonal lifted from S3
ρ8202-1-1-122-12-1-12-1-1-1-1-12-12-100-1-12-12-1    orthogonal lifted from S3
ρ920222222-1-122-1-1-1-1-1-1-1-1-1-100-12-122-1    orthogonal lifted from S3
ρ102022222222222222222-1-1-100-1-1-1-1-1-1    orthogonal lifted from S3
ρ11202-1-1-1222-1-1-1-122-1-1-1-12-1-100-122-1-1-1    orthogonal lifted from S3
ρ12202-1-1-122-12-1-12-1-1-1-1-12-1-120022-1-1-1-1    orthogonal lifted from S3
ρ1320222222-1-122-1-1-1-1-1-1-1-1-1-1002-12-1-12    orthogonal lifted from S3
ρ1420222222-1-122-1-1-1-1-1-1-122200-1-1-1-1-1-1    orthogonal lifted from S3
ρ15202-1-1-1222-1-1-1-122-1-1-1-1-1-1200-1-1-1-122    orthogonal lifted from S3
ρ16313333-3-3√-3/2-3+3√-3/200-3+3√-3/2-3-3√-3/20000000000ζ32ζ3000000    complex lifted from He3⋊C2
ρ173-13333-3+3√-3/2-3-3√-3/200-3-3√-3/2-3+3√-3/20000000000ζ65ζ6000000    complex lifted from He3⋊C2
ρ183-13333-3-3√-3/2-3+3√-3/200-3+3√-3/2-3-3√-3/20000000000ζ6ζ65000000    complex lifted from He3⋊C2
ρ19313333-3+3√-3/2-3-3√-3/200-3-3√-3/2-3+3√-3/20000000000ζ3ζ32000000    complex lifted from He3⋊C2
ρ2060-36-3-3000-30003-30-33300000000000    orthogonal lifted from C33⋊S3
ρ2160-3-3-36003-3000-30-330300000000000    orthogonal lifted from C33⋊S3
ρ2260-3-3-36000300-33-330-3000000000000    orthogonal lifted from C33⋊S3
ρ2360-3-36-300-3-30000330-3300000000000    orthogonal lifted from C33⋊S3
ρ2460-36-3-30030003-3030-3-300000000000    orthogonal lifted from C33⋊S3
ρ2560-3-36-300000033-3-330-300000000000    orthogonal lifted from C33⋊S3
ρ2660-3-36-3003300-3-300-33000000000000    orthogonal lifted from C33⋊S3
ρ2760-36-3-300-3300-303-330000000000000    orthogonal lifted from C33⋊S3
ρ2860-3-3-3600-30003030-33-300000000000    orthogonal lifted from C33⋊S3
ρ29606-3-3-3-3-3√-3-3+3√-3003-3√-3/23+3√-3/2000000000000000000    complex lifted from He3⋊5S3
ρ30606-3-3-3-3+3√-3-3-3√-3003+3√-3/23-3√-3/2000000000000000000    complex lifted from He3⋊5S3

Permutation representations of C34⋊7S3
►On 27 points - transitive group 27T165
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 8 26)(2 9 27)(3 7 25)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)

(1 8 26)(2 9 27)(3 7 25)(4 19 23)(5 20 24)(6 21 22)

(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)

(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), (1,8,26)(2,9,27)(3,7,25)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (1,8,26)(2,9,27)(3,7,25)(4,19,23)(5,20,24)(6,21,22), (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), (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), (1,8,26)(2,9,27)(3,7,25)(4,19,23)(5,20,24)(6,21,22)(10,18,15)(11,16,13)(12,17,14), (1,8,26)(2,9,27)(3,7,25)(4,19,23)(5,20,24)(6,21,22), (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), (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)], [(1,8,26),(2,9,27),(3,7,25),(4,19,23),(5,20,24),(6,21,22),(10,18,15),(11,16,13),(12,17,14)], [(1,8,26),(2,9,27),(3,7,25),(4,19,23),(5,20,24),(6,21,22)], [(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)], [(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(27,165);
 

Matrix representation of C34⋊7S3 ►in GL8(ℤ)

0-1000000
1-1000000
00010000
00-1-10000
00000100
0000-1-100
00000001
000000-1-1
,
0-1000000
1-1000000
00-1-10000
00100000
00000100
0000-1-100
00000001
000000-1-1
,
10000000
01000000
00-1-10000
00100000
00001000
00000100
00000001
000000-1-1
,
10000000
01000000
00010000
00-1-10000
00000100
0000-1-100
00000001
000000-1-1
,
10000000
01000000
00000010
00000001
00100000
00010000
00001000
00000100
,
01000000
10000000
00110000
000-10000
00000011
0000000-1
00001100
00000-100

G:=sub<GL(8,Integers())| [0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0] >;
 

C34⋊7S3 in GAP, Magma, Sage, TeX 

C_3^4\rtimes_7S_3
 
% in TeX
 
G:=Group("C3^4:7S3");
 
// GroupNames label
 
G:=SmallGroup(486,185);
 
// by ID
 
G=gap.SmallGroup(486,185);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,218,867,303,11344,1096,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f=b^-1,e*c*e^-1=c*d=d*c,f*c*f=c*d^-1,d*e=e*d,f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

Export

Character table of C34⋊7S3 in TeX

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