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G = C33⋊2D9  order 486 = 2·35

2nd semidirect product of C33 and D9 acting via D9/C3=S3

non-abelian, supersoluble, monomial

Aliases: C33⋊2D9, C34.2S3, C32⋊C9⋊7S3, C33⋊C9⋊3C2, C32.7(C9⋊S3), C33.20(C3⋊S3), C3.6(C33⋊S3), C3.3(C32⋊2D9), C32.20(He3⋊C2), SmallGroup(486,52)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C33⋊C9 — C33⋊2D9
Chief series
C1 — C3 — C32 — C33 — C34 — C33⋊C9 — C33⋊2D9
Lower central
C33⋊C9 — C33⋊2D9
Upper central
C1

Generators and relations for C33⋊2D9
 G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, ac=ca, dad-1=ab-1c, eae=a-1bc-1, ebe=bc=cb, dbd-1=bc-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1456 in 144 conjugacy classes, 17 normal (8 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C33, C33, C33, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32⋊C9, C34, C32⋊D9, C3×C33⋊C2, C33⋊C9, C33⋊2D9
Quotients: C1, C2, S3, D9, C3⋊S3, C9⋊S3, He3⋊C2, C32⋊2D9, C33⋊S3, C33⋊2D9

Character table of C33⋊2D9

 class 123A3B3C3D3E3F3G3H3I3J3K3L3M3N3O3P3Q6A6B9A9B9C9D9E9F9G9H9I
 size 181222233666666666668181181818181818181818
ρ1111111111111111111111111111111    trivial
ρ21-111111111111111111-1-1111111111    linear of order 2
ρ320222222-1-1-1-1-1-1-12-1-1200-1-1-12-1-122-1    orthogonal lifted from S3
ρ420222222-1-1-1-1-1-1-12-1-12002-1-1-122-1-1-1    orthogonal lifted from S3
ρ5202222222222222222200-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ620222222-1-1-1-1-1-1-12-1-1200-122-1-1-1-1-12    orthogonal lifted from S3
ρ7202-1-1-122-1222-1-1-1-1-1-1-100ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ8202-1-1-122-1-1-1-1222-1-1-1-100ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ95+ζ94ζ95+ζ94    orthogonal lifted from D9
ρ9202-1-1-1222-1-1-1-1-1-1-122-100ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ97+ζ92ζ98+ζ9    orthogonal lifted from D9
ρ10202-1-1-1222-1-1-1-1-1-1-122-100ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ95+ζ94ζ97+ζ92    orthogonal lifted from D9
ρ11202-1-1-1222-1-1-1-1-1-1-122-100ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ98+ζ9ζ95+ζ94    orthogonal lifted from D9
ρ12202-1-1-122-1222-1-1-1-1-1-1-100ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ13202-1-1-122-1-1-1-1222-1-1-1-100ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ98+ζ9ζ98+ζ9    orthogonal lifted from D9
ρ14202-1-1-122-1-1-1-1222-1-1-1-100ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ97+ζ92ζ97+ζ92    orthogonal lifted from D9
ρ15202-1-1-122-1222-1-1-1-1-1-1-100ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ163-13333-3-3√-3/2-3+3√-3/20000000-3+3√-3/200-3-3√-3/2ζ6ζ65000000000    complex lifted from He3⋊C2
ρ173-13333-3+3√-3/2-3-3√-3/20000000-3-3√-3/200-3+3√-3/2ζ65ζ6000000000    complex lifted from He3⋊C2
ρ18313333-3-3√-3/2-3+3√-3/20000000-3+3√-3/200-3-3√-3/2ζ32ζ3000000000    complex lifted from He3⋊C2
ρ19313333-3+3√-3/2-3-3√-3/20000000-3-3√-3/200-3+3√-3/2ζ3ζ32000000000    complex lifted from He3⋊C2
ρ2060-3-36-300-33-3003-3003000000000000    orthogonal lifted from C33⋊S3
ρ2160-3-36-300303-3-3030-30000000000000    orthogonal lifted from C33⋊S3
ρ2260-3-36-3000-3033-3003-3000000000000    orthogonal lifted from C33⋊S3
ρ2360-36-3-300-3-303-303003000000000000    orthogonal lifted from C33⋊S3
ρ2460-36-3-300003-303-303-3000000000000    orthogonal lifted from C33⋊S3
ρ2560-3-3-36003-30303-30-30000000000000    orthogonal lifted from C33⋊S3
ρ2660-3-3-360003-30-30303-3000000000000    orthogonal lifted from C33⋊S3
ρ2760-36-3-30033-303-300-30000000000000    orthogonal lifted from C33⋊S3
ρ2860-3-3-3600-303-33-30003000000000000    orthogonal lifted from C33⋊S3
ρ29606-3-3-3-3-3√-3-3+3√-300000003-3√-3/2003+3√-3/200000000000    complex lifted from C32⋊2D9
ρ30606-3-3-3-3+3√-3-3-3√-300000003+3√-3/2003-3√-3/200000000000    complex lifted from C32⋊2D9

Permutation representations of C33⋊2D9
►On 27 points - transitive group 27T193
Generators in S27
(1 7 4)(2 12 22)(3 13 23)(5 15 25)(6 16 26)(8 18 19)(9 10 20)(11 17 14)(21 27 24)

(1 14 27)(2 19 15)(4 17 21)(5 22 18)(7 11 24)(8 25 12)

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

(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 9)(2 8)(3 7)(4 6)(10 21)(11 20)(12 19)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)
 
G:=sub<Sym(27)| (1,7,4)(2,12,22)(3,13,23)(5,15,25)(6,16,26)(8,18,19)(9,10,20)(11,17,14)(21,27,24), (1,14,27)(2,19,15)(4,17,21)(5,22,18)(7,11,24)(8,25,12), (1,27,14)(2,19,15)(3,20,16)(4,21,17)(5,22,18)(6,23,10)(7,24,11)(8,25,12)(9,26,13), (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), (1,9)(2,8)(3,7)(4,6)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)>;
 
G:=Group( (1,7,4)(2,12,22)(3,13,23)(5,15,25)(6,16,26)(8,18,19)(9,10,20)(11,17,14)(21,27,24), (1,14,27)(2,19,15)(4,17,21)(5,22,18)(7,11,24)(8,25,12), (1,27,14)(2,19,15)(3,20,16)(4,21,17)(5,22,18)(6,23,10)(7,24,11)(8,25,12)(9,26,13), (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), (1,9)(2,8)(3,7)(4,6)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22) );
 
G=PermutationGroup([[(1,7,4),(2,12,22),(3,13,23),(5,15,25),(6,16,26),(8,18,19),(9,10,20),(11,17,14),(21,27,24)], [(1,14,27),(2,19,15),(4,17,21),(5,22,18),(7,11,24),(8,25,12)], [(1,27,14),(2,19,15),(3,20,16),(4,21,17),(5,22,18),(6,23,10),(7,24,11),(8,25,12),(9,26,13)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,21),(11,20),(12,19),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22)]])
 
G:=TransitiveGroup(27,193);
 

Matrix representation of C33⋊2D9 ►in GL8(𝔽19)

10000000
01000000
000180000
001180000
00250100
00122181800
0014170001
00714001818
,
10000000
01000000
000180000
001180000
00251000
00250100
0014170001
00714001818
,
10000000
01000000
001810000
001800000
007170100
001714181800
001250001
0052001818
,
25000000
147000000
000018100
001016171800
005451710
005451701
0016917500
00151017500
,
147000000
25000000
000018100
00931200
0031021400
0041021400
001415142180
0021814211

G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,2,12,14,7,0,0,18,18,5,2,17,14,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,2,2,14,7,0,0,18,18,5,5,17,14,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,18,7,17,12,5,0,0,1,0,17,14,5,2,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[2,14,0,0,0,0,0,0,5,7,0,0,0,0,0,0,0,0,0,10,5,5,16,15,0,0,0,16,4,4,9,10,0,0,18,17,5,5,17,17,0,0,1,18,17,17,5,5,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[14,2,0,0,0,0,0,0,7,5,0,0,0,0,0,0,0,0,0,9,3,4,14,2,0,0,0,3,10,10,15,18,0,0,18,1,2,2,14,14,0,0,1,2,14,14,2,2,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,1] >;
 

C33⋊2D9 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_2D_9
 
% in TeX
 
G:=Group("C3^3:2D9");
 
// GroupNames label
 
G:=SmallGroup(486,52);
 
// by ID
 
G=gap.SmallGroup(486,52);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,265,223,218,548,867,11344,3250,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^9=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1*c,e*a*e=a^-1*b*c^-1,e*b*e=b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of C33⋊2D9 in TeX

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