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G = C33.S3  order 162 = 2·34

4th non-split extension by C33 of S3 acting faithfully

metabelian, supersoluble, monomial

Aliases: C33.4S3, 3- 1+2⋊3S3, C9⋊(C3×S3), C3⋊(C9⋊C6), C9⋊S3⋊4C3, (C3×C9)⋊5C6, C32.5(C3⋊S3), C32.18(C3×S3), (C3×3- 1+2)⋊2C2, C3.3(C3×C3⋊S3), SmallGroup(162,42)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C9 — C33.S3
Chief series
C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — C33.S3
Lower central
C3×C9 — C33.S3
Upper central
C1

Generators and relations for C33.S3
 G = < a,b,c,d,e | a3=b3=c3=e2=1, d3=c, ab=ba, ac=ca, dad-1=ac-1, ae=ea, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=c-1d2 >

C1
27C2
C3
C3
C3
C3
3C3
6C3
9S3
9S3
9S3
9S3
27C6
C9
C9
C32
C9
C32
2C9
2C9
2C9
2C32
3C32
3C32
3C32
3D9
3D9
3C3⋊S3
3D9
9C3×S3
9C3×S3
9C3×S3
9C3×S3
C3×C9
C33
3- 1+2
3- 1+2
3- 1+2
2C3×C9
23- 1+2
23- 1+2
23- 1+2
C9⋊S3
3C9⋊C6
3C9⋊C6
3C3×C3⋊S3
3C9⋊C6
C3×3- 1+2
C33.S3

Character table of C33.S3

 class 123A3B3C3D3E3F3G3H6A6B9A9B9C9D9E9F9G9H9I
 size 127222233662727666666666
ρ1111111111111111111111    trivial
ρ21-111111111-1-1111111111    linear of order 2
ρ31-11111ζ32ζ3ζ32ζ3ζ6ζ65ζ311ζ32ζ32ζ321ζ3ζ3    linear of order 6
ρ4111111ζ32ζ3ζ32ζ3ζ32ζ3ζ311ζ32ζ32ζ321ζ3ζ3    linear of order 3
ρ51-11111ζ3ζ32ζ3ζ32ζ65ζ6ζ3211ζ3ζ3ζ31ζ32ζ32    linear of order 6
ρ6111111ζ3ζ32ζ3ζ32ζ3ζ32ζ3211ζ3ζ3ζ31ζ32ζ32    linear of order 3
ρ7202222222200-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ820-1-1-1222-1-100-1-1-1-12-12-12    orthogonal lifted from S3
ρ920-1-1-1222-1-100-1-122-1-1-12-1    orthogonal lifted from S3
ρ1020-1-1-1222-1-10022-1-1-12-1-1-1    orthogonal lifted from S3
ρ11202222-1-√-3-1+√-3-1-√-3-1+√-300ζ65-1-1ζ6ζ6ζ6-1ζ65ζ65    complex lifted from C3×S3
ρ12202222-1+√-3-1-√-3-1+√-3-1-√-300ζ6-1-1ζ65ζ65ζ65-1ζ6ζ6    complex lifted from C3×S3
ρ1320-1-1-12-1+√-3-1-√-3ζ65ζ600ζ6-12-1+√-3ζ65ζ65-1-1-√-3ζ6    complex lifted from C3×S3
ρ1420-1-1-12-1-√-3-1+√-3ζ6ζ6500-1+√-32-1ζ6ζ6-1-√-3-1ζ65ζ65    complex lifted from C3×S3
ρ1520-1-1-12-1-√-3-1+√-3ζ6ζ6500ζ65-1-1ζ6-1-√-3ζ62ζ65-1+√-3    complex lifted from C3×S3
ρ1620-1-1-12-1-√-3-1+√-3ζ6ζ6500ζ65-12-1-√-3ζ6ζ6-1-1+√-3ζ65    complex lifted from C3×S3
ρ1720-1-1-12-1+√-3-1-√-3ζ65ζ600-1-√-32-1ζ65ζ65-1+√-3-1ζ6ζ6    complex lifted from C3×S3
ρ1820-1-1-12-1+√-3-1-√-3ζ65ζ600ζ6-1-1ζ65-1+√-3ζ652ζ6-1-√-3    complex lifted from C3×S3
ρ19606-3-3-3000000000000000    orthogonal lifted from C9⋊C6
ρ2060-36-3-3000000000000000    orthogonal lifted from C9⋊C6
ρ2160-3-36-3000000000000000    orthogonal lifted from C9⋊C6

Permutation representations of C33.S3
►On 27 points - transitive group 27T58
Generators in S27
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(20 26 23)(21 24 27)

(1 14 25)(2 15 26)(3 16 27)(4 17 19)(5 18 20)(6 10 21)(7 11 22)(8 12 23)(9 13 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 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 20)(11 19)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)
 
G:=sub<Sym(27)| (2,8,5)(3,6,9)(10,13,16)(12,18,15)(20,26,23)(21,24,27), (1,14,25)(2,15,26)(3,16,27)(4,17,19)(5,18,20)(6,10,21)(7,11,22)(8,12,23)(9,13,24), (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), (1,2,3,4,5,6,7,8,9)(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)(10,20)(11,19)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)>;
 
G:=Group( (2,8,5)(3,6,9)(10,13,16)(12,18,15)(20,26,23)(21,24,27), (1,14,25)(2,15,26)(3,16,27)(4,17,19)(5,18,20)(6,10,21)(7,11,22)(8,12,23)(9,13,24), (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), (1,2,3,4,5,6,7,8,9)(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)(10,20)(11,19)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21) );
 
G=PermutationGroup([[(2,8,5),(3,6,9),(10,13,16),(12,18,15),(20,26,23),(21,24,27)], [(1,14,25),(2,15,26),(3,16,27),(4,17,19),(5,18,20),(6,10,21),(7,11,22),(8,12,23),(9,13,24)], [(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)], [(1,2,3,4,5,6,7,8,9),(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),(10,20),(11,19),(12,27),(13,26),(14,25),(15,24),(16,23),(17,22),(18,21)]])
 
G:=TransitiveGroup(27,58);
 

C33.S3 is a maximal subgroup of
 S3×C9⋊C6  (C3×He3).S3  C33.(C3⋊S3)  C3.(C33⋊S3)  3- 1+2⋊D9  C34.S3  C9⋊S3⋊C32  He3.(C3×S3)  C92⋊9C6  C9⋊He3⋊2C2  C92⋊10C6  C92⋊11C6  C92⋊12C6  C34⋊7S3  He3.(C3⋊S3)  (C32×C9).S3  3- 1+4⋊C2  C34.11S3  C9○He3⋊3S3
C33.S3 is a maximal quotient of
 C33.Dic3  C33⋊D9  C9⋊(S3×C9)  C92⋊9C6  C34.7S3  C9⋊He3⋊2C2  C9⋊C9⋊2S3  C92⋊10C6  C92⋊11C6  C92⋊12C6  C34.11S3

Matrix representation of C33.S3 ►in GL8(𝔽19)

70000000
07000000
00100000
00010000
0011181800
00001000
00000001
0011001818
,
173000000
181000000
001810000
001800000
001800100
0001181800
001800001
0001001818
,
10000000
01000000
001810000
001800000
001800100
0001181800
001800001
0001001818
,
116000000
117000000
0011001817
000000118
000000018
001000018
000010018
000001018
,
216000000
117000000
00010000
00100000
00000001
00000010
00000100
00001000

G:=sub<GL(8,GF(19))| [7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,0,1,1,0,0,1,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[17,18,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,18,18,18,0,18,0,0,0,1,0,0,1,0,1,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,18,18,0,18,0,0,0,1,0,0,1,0,1,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,1,0,0,0,0,0,0,16,17,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,18,1,0,0,0,0,0,0,17,18,18,18,18,18],[2,1,0,0,0,0,0,0,16,17,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;
 

C33.S3 in GAP, Magma, Sage, TeX 

C_3^3.S_3
 
% in TeX
 
G:=Group("C3^3.S3");
 
// GroupNames label
 
G:=SmallGroup(162,42);
 
// by ID
 
G=gap.SmallGroup(162,42);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-3,-3,992,457,282,723,2704]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^3=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*c^-1,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=c^-1*d^2>;
 
// generators/relations
 

Export

Subgroup lattice of C33.S3 in TeX
Character table of C33.S3 in TeX

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