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G = C32⋊D18  order 324 = 22·34

The semidirect product of C32 and D18 acting via D18/C3=D6

non-abelian, supersoluble, monomial

Aliases: C32⋊D18, C33.1D6, C3⋊S3⋊D9, C9⋊S3⋊1S3, (C3×C9)⋊1D6, C32⋊C18⋊C2, C32.7S32, C3.3(S3×D9), C32⋊C9⋊C22, C32⋊2D9⋊C2, C32⋊D9⋊C2, C3.1(C32⋊D6), (C3×C3⋊S3).S3, SmallGroup(324,37)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C32⋊C9 — C32⋊D18
Chief series
C1 — C3 — C32 — C33 — C32⋊C9 — C32⋊C18 — C32⋊D18
Lower central
C32⋊C9 — C32⋊D18
Upper central
C1

Generators and relations for C32⋊D18
 G = < a,b,c,d | a3=b3=c18=d2=1, ab=ba, cac-1=dad=a-1b, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 597 in 67 conjugacy classes, 15 normal (all characteristic)
C1, C2, C3, C3, C22, S3, C6, C9, C32, C32, D6, D9, C18, C3×S3, C3⋊S3, C3⋊S3, C3×C9, C3×C9, C33, D18, S32, C3×D9, S3×C9, C9⋊S3, C3×C3⋊S3, C3×C3⋊S3, C32⋊C9, S3×D9, C32⋊4D6, C32⋊C18, C32⋊D9, C32⋊2D9, C32⋊D18
Quotients: C1, C2, C22, S3, D6, D9, D18, S32, S3×D9, C32⋊D6, C32⋊D18

Character table of C32⋊D18

 class 12A2B2C3A3B3C3D3E6A6B6C9A9B9C9D9E9F18A18B18C
 size 192727224612185454666121212181818
ρ1111111111111111111111    trivial
ρ21-1-1111111-1-11111111-1-1-1    linear of order 2
ρ31-11-111111-11-1111111-1-1-1    linear of order 2
ρ411-1-1111111-1-1111111111    linear of order 2
ρ52-20022222-200-1-1-1-1-1-1111    orthogonal lifted from D6
ρ6220022222200-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ72002222-1-100-1222-1-1-1000    orthogonal lifted from S3
ρ8200-2222-1-1001222-1-1-1000    orthogonal lifted from D6
ρ92200-12-12-1-100ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ95+ζ94ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ102-200-12-12-1100ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ95+ζ94ζ97+ζ92-ζ98-ζ9-ζ97-ζ92-ζ95-ζ94    orthogonal lifted from D18
ρ112-200-12-12-1100ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ97+ζ92ζ98+ζ9-ζ95-ζ94-ζ98-ζ9-ζ97-ζ92    orthogonal lifted from D18
ρ122200-12-12-1-100ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ98+ζ9ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ132200-12-12-1-100ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ97+ζ92ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ142-200-12-12-1100ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ98+ζ9ζ95+ζ94-ζ97-ζ92-ζ95-ζ94-ζ98-ζ9    orthogonal lifted from D18
ρ154000444-2-2000-2-2-2111000    orthogonal lifted from S32
ρ164000-24-2-210002ζ95+2ζ942ζ98+2ζ92ζ97+2ζ92-ζ97-ζ92-ζ98-ζ9-ζ95-ζ94000    orthogonal lifted from S3×D9
ρ174000-24-2-210002ζ98+2ζ92ζ97+2ζ922ζ95+2ζ94-ζ95-ζ94-ζ97-ζ92-ζ98-ζ9000    orthogonal lifted from S3×D9
ρ184000-24-2-210002ζ97+2ζ922ζ95+2ζ942ζ98+2ζ9-ζ98-ζ9-ζ95-ζ94-ζ97-ζ92000    orthogonal lifted from S3×D9
ρ1960-206-3-300010000000000    orthogonal lifted from C32⋊D6
ρ2060206-3-3000-10000000000    orthogonal lifted from C32⋊D6
ρ2112000-6-6300000000000000    orthogonal faithful

Permutation representations of C32⋊D18
►On 18 points - transitive group 18T132
Generators in S18
(2 14 8)(3 15 9)(5 11 17)(6 12 18)

(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)

(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)
 
G:=sub<Sym(18)| (2,14,8)(3,15,9)(5,11,17)(6,12,18), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)>;
 
G:=Group( (2,14,8)(3,15,9)(5,11,17)(6,12,18), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10) );
 
G=PermutationGroup([[(2,14,8),(3,15,9),(5,11,17),(6,12,18)], [(1,13,7),(2,8,14),(3,15,9),(4,10,16),(5,17,11),(6,12,18)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]])
 
G:=TransitiveGroup(18,132);
 

►On 27 points - transitive group 27T126
Generators in S27
(1 19 10)(2 20 11)(4 13 22)(5 14 23)(7 25 16)(8 26 17)

(1 10 19)(2 20 11)(3 12 21)(4 22 13)(5 14 23)(6 24 15)(7 16 25)(8 26 17)(9 18 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 7)(2 6)(3 5)(8 9)(10 16)(11 15)(12 14)(17 27)(18 26)(19 25)(20 24)(21 23)
 
G:=sub<Sym(27)| (1,19,10)(2,20,11)(4,13,22)(5,14,23)(7,25,16)(8,26,17), (1,10,19)(2,20,11)(3,12,21)(4,22,13)(5,14,23)(6,24,15)(7,16,25)(8,26,17)(9,18,27), (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), (1,7)(2,6)(3,5)(8,9)(10,16)(11,15)(12,14)(17,27)(18,26)(19,25)(20,24)(21,23)>;
 
G:=Group( (1,19,10)(2,20,11)(4,13,22)(5,14,23)(7,25,16)(8,26,17), (1,10,19)(2,20,11)(3,12,21)(4,22,13)(5,14,23)(6,24,15)(7,16,25)(8,26,17)(9,18,27), (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), (1,7)(2,6)(3,5)(8,9)(10,16)(11,15)(12,14)(17,27)(18,26)(19,25)(20,24)(21,23) );
 
G=PermutationGroup([[(1,19,10),(2,20,11),(4,13,22),(5,14,23),(7,25,16),(8,26,17)], [(1,10,19),(2,20,11),(3,12,21),(4,22,13),(5,14,23),(6,24,15),(7,16,25),(8,26,17),(9,18,27)], [(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)], [(1,7),(2,6),(3,5),(8,9),(10,16),(11,15),(12,14),(17,27),(18,26),(19,25),(20,24),(21,23)]])
 
G:=TransitiveGroup(27,126);
 

Matrix representation of C32⋊D18 ►in GL10(𝔽19)

00180000000
00018000000
10180000000
01018000000
0000000010
0000100000
0000000100
0000000001
0000010000
0000001000
,
1000000000
0100000000
0010000000
0001000000
00000018000
00000000018
00001018000
00000001810
00000001800
00000100018
,
001111000000
00815000000
111100000000
81500000000
0000001000
00000000180
0000100000
00000000018
00000100018
00000001180
,
001111000000
00158000000
111100000000
15800000000
0000100000
00000001180
0000001000
00000100018
00000000018
00000000180

G:=sub<GL(10,GF(19))| [0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,18,0,0,0,0,0,0,0,0,18,0,18,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,18,0,18,0,0,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,18],[0,0,11,8,0,0,0,0,0,0,0,0,11,15,0,0,0,0,0,0,11,8,0,0,0,0,0,0,0,0,11,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,18,0,0,0,0,0,0,0,18,18,0],[0,0,11,15,0,0,0,0,0,0,0,0,11,8,0,0,0,0,0,0,11,15,0,0,0,0,0,0,0,0,11,8,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,0,0,0,18,0,0,0,0,0,0,0,18,18,0] >;
 

C32⋊D18 in GAP, Magma, Sage, TeX 

C_3^2\rtimes D_{18}
 
% in TeX
 
G:=Group("C3^2:D18");
 
// GroupNames label
 
G:=SmallGroup(324,37);
 
// by ID
 
G=gap.SmallGroup(324,37);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,404,338,579,735,1090,7781,3899]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^18=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C32⋊D18 in TeX

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