Copied to
clipboard

G = C32⋊D9order 162 = 2·34

1st semidirect product of C32 and D9 acting via D9/C3=S3

metabelian, supersoluble, monomial

Aliases: C321D9, C33.1S3, C9⋊S31C3, C3.(C9⋊C6), (C3×C9)⋊1C6, C32⋊C92C2, C3.1(C3×D9), C3.1(C32⋊C6), C32.13(C3×S3), SmallGroup(162,5)

Series: Derived Chief Lower central Upper central

C1C3×C9 — C32⋊D9
C1C3C32C3×C9C32⋊C9 — C32⋊D9
C3×C9 — C32⋊D9
C1

Generators and relations for C32⋊D9
 G = < a,b,c,d | a3=b3=c9=d2=1, cac-1=dad=ab=ba, bc=cb, dbd=b-1, dcd=c-1 >

27C2
3C3
6C3
9S3
9S3
9S3
9S3
27C6
2C32
3C32
3C32
3C9
3C32
6C9
3C3⋊S3
9C3×S3
9D9
9C3×S3
9C3×S3
9C3×S3
2C3×C9
3C3×C3⋊S3

Character table of C32⋊D9

 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
ρ8202-1-1-122-1-100ζ9792ζ9792ζ9594ζ9594ζ989ζ9792ζ989ζ9594ζ989    orthogonal lifted from D9
ρ9202-1-1-122-1-100ζ9594ζ9594ζ989ζ989ζ9792ζ9594ζ9792ζ989ζ9792    orthogonal lifted from D9
ρ10202-1-1-122-1-100ζ989ζ989ζ9792ζ9792ζ9594ζ989ζ9594ζ9792ζ9594    orthogonal lifted from D9
ρ11202222-1+-3-1--3-1+-3-1--300ζ6-1-1ζ65ζ65ζ65-1ζ6ζ6    complex lifted from C3×S3
ρ12202-1-1-1-1--3-1+-3ζ6ζ6500ζ959ζ9792ζ9594ζ929ζ9795ζ9894ζ989ζ9897ζ9492    complex lifted from C3×D9
ρ13202222-1--3-1+-3-1--3-1+-300ζ65-1-1ζ6ζ6ζ6-1ζ65ζ65    complex lifted from C3×S3
ρ14202-1-1-1-1+-3-1--3ζ65ζ600ζ9894ζ9792ζ9594ζ9897ζ9492ζ959ζ989ζ929ζ9795    complex lifted from C3×D9
ρ15202-1-1-1-1+-3-1--3ζ65ζ600ζ9795ζ989ζ9792ζ959ζ9897ζ9492ζ9594ζ9894ζ929    complex lifted from C3×D9
ρ16202-1-1-1-1--3-1+-3ζ6ζ6500ζ9897ζ9594ζ989ζ9795ζ9894ζ929ζ9792ζ9492ζ959    complex lifted from C3×D9
ρ17202-1-1-1-1--3-1+-3ζ6ζ6500ζ9492ζ989ζ9792ζ9894ζ929ζ9795ζ9594ζ959ζ9897    complex lifted from C3×D9
ρ18202-1-1-1-1+-3-1--3ζ65ζ600ζ929ζ9594ζ989ζ9492ζ959ζ9897ζ9792ζ9795ζ9894    complex lifted from C3×D9
ρ1960-36-3-3000000000000000    orthogonal lifted from C9⋊C6
ρ2060-3-3-36000000000000000    orthogonal lifted from C32⋊C6
ρ2160-3-36-3000000000000000    orthogonal lifted from C9⋊C6

Permutation representations of C32⋊D9
On 27 points - transitive group 27T59
Generators in S27
(2 26 15)(3 16 27)(5 20 18)(6 10 21)(8 23 12)(9 13 24)
(1 25 14)(2 26 15)(3 27 16)(4 19 17)(5 20 18)(6 21 10)(7 22 11)(8 23 12)(9 24 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 19)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)

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

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

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

G:=TransitiveGroup(27,59);

C32⋊D9 is a maximal subgroup of
C32⋊D18  (C3×He3)⋊S3  (C3×He3).S3  C33.(C3⋊S3)  C32⋊C96S3  C3.(C33⋊S3)  C3.(He3⋊S3)  C32⋊C9.10S3  C332D9  (C3×C9)⋊5D9  (C3×C9)⋊6D9  He32D9  3- 1+2⋊D9  C34.S3  C33⋊D9  C923C6  He33D9  C929C6  C9⋊He32C2  (C32×C9)⋊C6  C9210C6  C924C6  C925C6  C9211C6
C32⋊D9 is a maximal quotient of
C32⋊Dic9  C9⋊S3⋊C9  C32⋊D27  C331D9  (C3×C9)⋊D9  (C3×C9)⋊3D9  He3⋊D9  He3.D9  He3.2D9  C33⋊D9

Matrix representation of C32⋊D9 in GL8(𝔽19)

70000000
07000000
00100000
00010000
0000181800
00001000
00000001
0000001818
,
10000000
01000000
00010000
0018180000
00000100
0000181800
00000001
0000001818
,
214000000
57000000
00001000
00000100
00000010
00000001
0018180000
00100000
,
214000000
1217000000
00001000
0000181800
00100000
0018180000
00000001
00000010

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

C32⋊D9 in GAP, Magma, Sage, TeX

C_3^2\rtimes D_9
% in TeX

G:=Group("C3^2:D9");
// GroupNames label

G:=SmallGroup(162,5);
// by ID

G=gap.SmallGroup(162,5);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,992,187,282,723,2704]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^9=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of C32⋊D9 in TeX
Character table of C32⋊D9 in TeX

׿
×
𝔽