Copied to
clipboard

## G = C32⋊C18order 162 = 2·34

### The semidirect product of C32 and C18 acting via C18/C3=C6

Aliases: C32⋊C18, C33.1C6, C3⋊S3⋊C9, (C3×C9)⋊1S3, C3.2(S3×C9), C32⋊C91C2, C3.5(C32⋊C6), C32.12(C3×S3), (C3×C3⋊S3).C3, SmallGroup(162,4)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C32⋊C18
G = < a,b,c | a3=b3=c18=1, ab=ba, cac-1=a-1b-1, cbc-1=b-1 >

Character table of C32⋊C18

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 9A 9B 9C 9D 9E 9F 9G 9H 9I 9J 9K 9L 18A 18B 18C 18D 18E 18F size 1 9 1 1 2 2 2 6 6 6 9 9 3 3 3 3 3 3 6 6 6 6 6 6 9 9 9 9 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ4 1 -1 1 1 1 1 1 1 1 1 -1 -1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ5 1 -1 1 1 1 1 1 1 1 1 -1 -1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ6 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ7 1 -1 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ65 ζ95 ζ98 ζ92 ζ9 ζ94 ζ97 ζ98 ζ94 ζ92 ζ9 ζ95 ζ97 -ζ92 -ζ9 -ζ94 -ζ95 -ζ98 -ζ97 linear of order 18 ρ8 1 1 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ97 ζ94 ζ9 ζ95 ζ92 ζ98 ζ94 ζ92 ζ9 ζ95 ζ97 ζ98 ζ9 ζ95 ζ92 ζ97 ζ94 ζ98 linear of order 9 ρ9 1 -1 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ6 ζ97 ζ94 ζ9 ζ95 ζ92 ζ98 ζ94 ζ92 ζ9 ζ95 ζ97 ζ98 -ζ9 -ζ95 -ζ92 -ζ97 -ζ94 -ζ98 linear of order 18 ρ10 1 -1 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ65 ζ92 ζ95 ζ98 ζ94 ζ97 ζ9 ζ95 ζ97 ζ98 ζ94 ζ92 ζ9 -ζ98 -ζ94 -ζ97 -ζ92 -ζ95 -ζ9 linear of order 18 ρ11 1 -1 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ6 ζ94 ζ9 ζ97 ζ98 ζ95 ζ92 ζ9 ζ95 ζ97 ζ98 ζ94 ζ92 -ζ97 -ζ98 -ζ95 -ζ94 -ζ9 -ζ92 linear of order 18 ρ12 1 1 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ95 ζ98 ζ92 ζ9 ζ94 ζ97 ζ98 ζ94 ζ92 ζ9 ζ95 ζ97 ζ92 ζ9 ζ94 ζ95 ζ98 ζ97 linear of order 9 ρ13 1 1 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ94 ζ9 ζ97 ζ98 ζ95 ζ92 ζ9 ζ95 ζ97 ζ98 ζ94 ζ92 ζ97 ζ98 ζ95 ζ94 ζ9 ζ92 linear of order 9 ρ14 1 -1 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ6 ζ9 ζ97 ζ94 ζ92 ζ98 ζ95 ζ97 ζ98 ζ94 ζ92 ζ9 ζ95 -ζ94 -ζ92 -ζ98 -ζ9 -ζ97 -ζ95 linear of order 18 ρ15 1 1 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ98 ζ92 ζ95 ζ97 ζ9 ζ94 ζ92 ζ9 ζ95 ζ97 ζ98 ζ94 ζ95 ζ97 ζ9 ζ98 ζ92 ζ94 linear of order 9 ρ16 1 -1 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ65 ζ98 ζ92 ζ95 ζ97 ζ9 ζ94 ζ92 ζ9 ζ95 ζ97 ζ98 ζ94 -ζ95 -ζ97 -ζ9 -ζ98 -ζ92 -ζ94 linear of order 18 ρ17 1 1 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ92 ζ95 ζ98 ζ94 ζ97 ζ9 ζ95 ζ97 ζ98 ζ94 ζ92 ζ9 ζ98 ζ94 ζ97 ζ92 ζ95 ζ9 linear of order 9 ρ18 1 1 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ9 ζ97 ζ94 ζ92 ζ98 ζ95 ζ97 ζ98 ζ94 ζ92 ζ9 ζ95 ζ94 ζ92 ζ98 ζ9 ζ97 ζ95 linear of order 9 ρ19 2 0 2 2 2 2 2 -1 -1 -1 0 0 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from S3 ρ20 2 0 2 2 2 2 2 -1 -1 -1 0 0 -1-√-3 -1-√-3 -1-√-3 -1+√-3 -1+√-3 -1+√-3 ζ6 ζ65 ζ6 ζ65 ζ6 ζ65 0 0 0 0 0 0 complex lifted from C3×S3 ρ21 2 0 2 2 2 2 2 -1 -1 -1 0 0 -1+√-3 -1+√-3 -1+√-3 -1-√-3 -1-√-3 -1-√-3 ζ65 ζ6 ζ65 ζ6 ζ65 ζ6 0 0 0 0 0 0 complex lifted from C3×S3 ρ22 2 0 -1+√-3 -1-√-3 2 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 2ζ9 2ζ97 2ζ94 2ζ92 2ζ98 2ζ95 -ζ97 -ζ98 -ζ94 -ζ92 -ζ9 -ζ95 0 0 0 0 0 0 complex lifted from S3×C9 ρ23 2 0 -1-√-3 -1+√-3 2 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 2ζ98 2ζ92 2ζ95 2ζ97 2ζ9 2ζ94 -ζ92 -ζ9 -ζ95 -ζ97 -ζ98 -ζ94 0 0 0 0 0 0 complex lifted from S3×C9 ρ24 2 0 -1+√-3 -1-√-3 2 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 2ζ97 2ζ94 2ζ9 2ζ95 2ζ92 2ζ98 -ζ94 -ζ92 -ζ9 -ζ95 -ζ97 -ζ98 0 0 0 0 0 0 complex lifted from S3×C9 ρ25 2 0 -1+√-3 -1-√-3 2 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 2ζ94 2ζ9 2ζ97 2ζ98 2ζ95 2ζ92 -ζ9 -ζ95 -ζ97 -ζ98 -ζ94 -ζ92 0 0 0 0 0 0 complex lifted from S3×C9 ρ26 2 0 -1-√-3 -1+√-3 2 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 2ζ95 2ζ98 2ζ92 2ζ9 2ζ94 2ζ97 -ζ98 -ζ94 -ζ92 -ζ9 -ζ95 -ζ97 0 0 0 0 0 0 complex lifted from S3×C9 ρ27 2 0 -1-√-3 -1+√-3 2 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 2ζ92 2ζ95 2ζ98 2ζ94 2ζ97 2ζ9 -ζ95 -ζ97 -ζ98 -ζ94 -ζ92 -ζ9 0 0 0 0 0 0 complex lifted from S3×C9 ρ28 6 0 6 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ29 6 0 -3-3√-3 -3+3√-3 -3 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ30 6 0 -3+3√-3 -3-3√-3 -3 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C32⋊C18
On 18 points - transitive group 18T82
Generators in S18
```(2 8 14)(3 9 15)(5 17 11)(6 18 12)
(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)```

`G:=sub<Sym(18)| (2,8,14)(3,9,15)(5,17,11)(6,18,12), (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)>;`

`G:=Group( (2,8,14)(3,9,15)(5,17,11)(6,18,12), (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) );`

`G=PermutationGroup([[(2,8,14),(3,9,15),(5,17,11),(6,18,12)], [(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)]])`

`G:=TransitiveGroup(18,82);`

On 27 points - transitive group 27T47
Generators in S27
```(1 16 25)(3 27 18)(4 10 19)(6 21 12)(7 22 13)(9 15 24)
(1 25 16)(2 17 26)(3 27 18)(4 19 10)(5 11 20)(6 21 12)(7 13 22)(8 23 14)(9 15 24)
(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)```

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

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

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

`G:=TransitiveGroup(27,47);`

C32⋊C18 is a maximal subgroup of
C32⋊D18  C32⋊C9.S3  C32⋊C9⋊S3  (C3×He3).C6  C32⋊C9.C6  C33.(C3×S3)  C322D9.C3  C331C18  (C3×C9)⋊C18  C9⋊S33C9  C9×C32⋊C6  C34.C6  C9⋊He3⋊C2  C33⋊C18  C923S3  (C32×C9)⋊8S3  C926S3  C925S3
C32⋊C18 is a maximal quotient of
C32⋊C36  C9⋊S3⋊C9  C32⋊C54  C331C18  (C3×C9)⋊C18  C9⋊S33C9  He3⋊C18  He3.C18  He3.2C18  C33⋊C18

Matrix representation of C32⋊C18 in GL6(𝔽19)

 1 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 12 0 7 12 1 7
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 8 18 12 0 0 11
,
 18 12 8 18 12 9 0 0 0 7 0 0 0 0 0 0 7 0 0 0 11 0 0 0 7 0 0 0 0 0 18 12 0 18 12 1

`G:=sub<GL(6,GF(19))| [1,0,0,0,0,12,0,7,0,0,0,0,0,0,11,0,0,7,0,0,0,1,0,12,0,0,0,0,11,1,0,0,0,0,0,7],[7,0,0,0,0,8,0,7,0,0,0,18,0,0,7,0,0,12,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[18,0,0,0,7,18,12,0,0,0,0,12,8,0,0,11,0,0,18,7,0,0,0,18,12,0,7,0,0,12,9,0,0,0,0,1] >;`

C32⋊C18 in GAP, Magma, Sage, TeX

`C_3^2\rtimes C_{18}`
`% in TeX`

`G:=Group("C3^2:C18");`
`// GroupNames label`

`G:=SmallGroup(162,4);`
`// by ID`

`G=gap.SmallGroup(162,4);`
`# by ID`

`G:=PCGroup([5,-2,-3,-3,-3,-3,36,723,728,2704]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=b^3=c^18=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=b^-1>;`
`// generators/relations`

Export

׿
×
𝔽