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G = C32⋊D18order 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⋊S31S3, (C3×C9)⋊1D6, C32⋊C18⋊C2, C32.7S32, C3.3(S3×D9), C32⋊C9⋊C22, C322D9⋊C2, C32⋊D9⋊C2, C3.1(C32⋊D6), (C3×C3⋊S3).S3, SmallGroup(324,37)

Series: Derived Chief Lower central Upper central

C1C3C32⋊C9 — C32⋊D18
C1C3C32C33C32⋊C9C32⋊C18 — C32⋊D18
C32⋊C9 — C32⋊D18
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 [×3], C3 [×2], C3 [×3], C22, S3 [×8], C6 [×3], C9 [×2], C32 [×2], C32 [×3], D6 [×3], D9 [×3], C18, C3×S3 [×8], C3⋊S3, C3⋊S3 [×2], C3×C9, C3×C9, C33, D18, S32 [×3], C3×D9 [×2], S3×C9, C9⋊S3, C3×C3⋊S3, C3×C3⋊S3 [×2], C32⋊C9, S3×D9, C324D6, C32⋊C18, C32⋊D9, C322D9, C32⋊D18
Quotients: C1, C2 [×3], C22, S3 [×2], D6 [×2], 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
ρ92200-12-12-1-100ζ9792ζ9594ζ989ζ989ζ9594ζ9792ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ102-200-12-12-1100ζ9792ζ9594ζ989ζ989ζ9594ζ979298997929594    orthogonal lifted from D18
ρ112-200-12-12-1100ζ989ζ9792ζ9594ζ9594ζ9792ζ98995949899792    orthogonal lifted from D18
ρ122200-12-12-1-100ζ9594ζ989ζ9792ζ9792ζ989ζ9594ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ132200-12-12-1-100ζ989ζ9792ζ9594ζ9594ζ9792ζ989ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ142-200-12-12-1100ζ9594ζ989ζ9792ζ9792ζ989ζ959497929594989    orthogonal lifted from D18
ρ154000444-2-2000-2-2-2111000    orthogonal lifted from S32
ρ164000-24-2-2100095+2ζ9498+2ζ997+2ζ9297929899594000    orthogonal lifted from S3×D9
ρ174000-24-2-2100098+2ζ997+2ζ9295+2ζ9495949792989000    orthogonal lifted from S3×D9
ρ184000-24-2-2100097+2ζ9295+2ζ9498+2ζ998995949792000    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 18 27)(2 19 10)(4 12 21)(5 13 22)(7 24 15)(8 25 16)
(1 27 18)(2 19 10)(3 11 20)(4 21 12)(5 13 22)(6 23 14)(7 15 24)(8 25 16)(9 17 26)
(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 4)(2 3)(5 9)(6 8)(10 20)(11 19)(12 18)(13 17)(14 16)(21 27)(22 26)(23 25)

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

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

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

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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