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## G = C42⋊2C18order 288 = 25·32

### 2nd semidirect product of C42 and C18 acting via C18/C3=C6

Aliases: C422C18, C41D4⋊C9, C42⋊C93C2, (C4×C12).2C6, (C22×C6).3A4, C3.(C23.A4), C23.2(C3.A4), (C2×C6).8(C2×A4), (C3×C41D4).C3, C22.4(C2×C3.A4), SmallGroup(288,75)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C42⋊2C18
 Chief series C1 — C22 — C42 — C4×C12 — C42⋊C9 — C42⋊2C18
 Lower central C42 — C42⋊2C18
 Upper central C1 — C3

Generators and relations for C422C18
G = < a,b,c | a4=b4=c18=1, ab=ba, cac-1=a2b-1, cbc-1=a-1b-1 >

Character table of C422C18

 class 1 2A 2B 2C 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 18A 18B 18C 18D 18E 18F size 1 3 4 12 1 1 6 6 3 3 4 4 12 12 16 16 16 16 16 16 6 6 6 6 16 16 16 16 16 16 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 -1 -1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 ζ65 ζ6 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ6 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 ζ6 ζ65 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ7 1 1 -1 -1 ζ32 ζ3 1 1 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 ζ97 ζ92 ζ98 ζ94 ζ9 ζ95 ζ32 ζ32 ζ3 ζ3 -ζ9 -ζ95 -ζ92 -ζ97 -ζ94 -ζ98 linear of order 18 ρ8 1 1 -1 -1 ζ32 ζ3 1 1 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 ζ94 ζ95 ζ92 ζ9 ζ97 ζ98 ζ32 ζ32 ζ3 ζ3 -ζ97 -ζ98 -ζ95 -ζ94 -ζ9 -ζ92 linear of order 18 ρ9 1 1 -1 -1 ζ32 ζ3 1 1 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 ζ9 ζ98 ζ95 ζ97 ζ94 ζ92 ζ32 ζ32 ζ3 ζ3 -ζ94 -ζ92 -ζ98 -ζ9 -ζ97 -ζ95 linear of order 18 ρ10 1 1 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ92 ζ97 ζ9 ζ95 ζ98 ζ94 ζ3 ζ3 ζ32 ζ32 ζ98 ζ94 ζ97 ζ92 ζ95 ζ9 linear of order 9 ρ11 1 1 -1 -1 ζ3 ζ32 1 1 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 ζ95 ζ94 ζ97 ζ98 ζ92 ζ9 ζ3 ζ3 ζ32 ζ32 -ζ92 -ζ9 -ζ94 -ζ95 -ζ98 -ζ97 linear of order 18 ρ12 1 1 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ94 ζ95 ζ92 ζ9 ζ97 ζ98 ζ32 ζ32 ζ3 ζ3 ζ97 ζ98 ζ95 ζ94 ζ9 ζ92 linear of order 9 ρ13 1 1 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ9 ζ98 ζ95 ζ97 ζ94 ζ92 ζ32 ζ32 ζ3 ζ3 ζ94 ζ92 ζ98 ζ9 ζ97 ζ95 linear of order 9 ρ14 1 1 -1 -1 ζ3 ζ32 1 1 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 ζ92 ζ97 ζ9 ζ95 ζ98 ζ94 ζ3 ζ3 ζ32 ζ32 -ζ98 -ζ94 -ζ97 -ζ92 -ζ95 -ζ9 linear of order 18 ρ15 1 1 -1 -1 ζ3 ζ32 1 1 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 ζ98 ζ9 ζ94 ζ92 ζ95 ζ97 ζ3 ζ3 ζ32 ζ32 -ζ95 -ζ97 -ζ9 -ζ98 -ζ92 -ζ94 linear of order 18 ρ16 1 1 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ95 ζ94 ζ97 ζ98 ζ92 ζ9 ζ3 ζ3 ζ32 ζ32 ζ92 ζ9 ζ94 ζ95 ζ98 ζ97 linear of order 9 ρ17 1 1 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ98 ζ9 ζ94 ζ92 ζ95 ζ97 ζ3 ζ3 ζ32 ζ32 ζ95 ζ97 ζ9 ζ98 ζ92 ζ94 linear of order 9 ρ18 1 1 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ97 ζ92 ζ98 ζ94 ζ9 ζ95 ζ32 ζ32 ζ3 ζ3 ζ9 ζ95 ζ92 ζ97 ζ94 ζ98 linear of order 9 ρ19 3 3 3 -1 3 3 -1 -1 3 3 3 3 -1 -1 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ20 3 3 -3 1 3 3 -1 -1 3 3 -3 -3 1 1 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ21 3 3 -3 1 -3-3√-3/2 -3+3√-3/2 -1 -1 -3-3√-3/2 -3+3√-3/2 3-3√-3/2 3+3√-3/2 ζ32 ζ3 0 0 0 0 0 0 ζ6 ζ6 ζ65 ζ65 0 0 0 0 0 0 complex lifted from C2×C3.A4 ρ22 3 3 3 -1 -3+3√-3/2 -3-3√-3/2 -1 -1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 ζ65 ζ6 0 0 0 0 0 0 ζ65 ζ65 ζ6 ζ6 0 0 0 0 0 0 complex lifted from C3.A4 ρ23 3 3 3 -1 -3-3√-3/2 -3+3√-3/2 -1 -1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 ζ6 ζ65 0 0 0 0 0 0 ζ6 ζ6 ζ65 ζ65 0 0 0 0 0 0 complex lifted from C3.A4 ρ24 3 3 -3 1 -3+3√-3/2 -3-3√-3/2 -1 -1 -3+3√-3/2 -3-3√-3/2 3+3√-3/2 3-3√-3/2 ζ3 ζ32 0 0 0 0 0 0 ζ65 ζ65 ζ6 ζ6 0 0 0 0 0 0 complex lifted from C2×C3.A4 ρ25 6 -2 0 0 6 6 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 orthogonal lifted from C23.A4 ρ26 6 -2 0 0 6 6 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from C23.A4 ρ27 6 -2 0 0 -3-3√-3 -3+3√-3 -2 2 1+√-3 1-√-3 0 0 0 0 0 0 0 0 0 0 -1-√-3 1+√-3 -1+√-3 1-√-3 0 0 0 0 0 0 complex faithful ρ28 6 -2 0 0 -3-3√-3 -3+3√-3 2 -2 1+√-3 1-√-3 0 0 0 0 0 0 0 0 0 0 1+√-3 -1-√-3 1-√-3 -1+√-3 0 0 0 0 0 0 complex faithful ρ29 6 -2 0 0 -3+3√-3 -3-3√-3 2 -2 1-√-3 1+√-3 0 0 0 0 0 0 0 0 0 0 1-√-3 -1+√-3 1+√-3 -1-√-3 0 0 0 0 0 0 complex faithful ρ30 6 -2 0 0 -3+3√-3 -3-3√-3 -2 2 1-√-3 1+√-3 0 0 0 0 0 0 0 0 0 0 -1+√-3 1-√-3 -1-√-3 1+√-3 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C422C18
On 36 points
Generators in S36
```(2 11 30 21)(3 12 31 22)(5 24 33 14)(6 25 34 15)(8 17 36 27)(9 18 19 28)
(1 20 29 10)(2 11 30 21)(3 31)(4 13 32 23)(5 24 33 14)(6 34)(7 26 35 16)(8 17 36 27)(9 19)(12 22)(15 25)(18 28)
(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 28 29 30 31 32 33 34 35 36)```

`G:=sub<Sym(36)| (2,11,30,21)(3,12,31,22)(5,24,33,14)(6,25,34,15)(8,17,36,27)(9,18,19,28), (1,20,29,10)(2,11,30,21)(3,31)(4,13,32,23)(5,24,33,14)(6,34)(7,26,35,16)(8,17,36,27)(9,19)(12,22)(15,25)(18,28), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)>;`

`G:=Group( (2,11,30,21)(3,12,31,22)(5,24,33,14)(6,25,34,15)(8,17,36,27)(9,18,19,28), (1,20,29,10)(2,11,30,21)(3,31)(4,13,32,23)(5,24,33,14)(6,34)(7,26,35,16)(8,17,36,27)(9,19)(12,22)(15,25)(18,28), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36) );`

`G=PermutationGroup([[(2,11,30,21),(3,12,31,22),(5,24,33,14),(6,25,34,15),(8,17,36,27),(9,18,19,28)], [(1,20,29,10),(2,11,30,21),(3,31),(4,13,32,23),(5,24,33,14),(6,34),(7,26,35,16),(8,17,36,27),(9,19),(12,22),(15,25),(18,28)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)]])`

Matrix representation of C422C18 in GL6(𝔽37)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 9 0 36 0 0 0 7 0 0 0 0 36 0 0 0 0 1 0
,
 1 35 0 0 0 0 1 36 0 0 0 0 0 28 0 1 0 0 9 28 36 0 0 0 7 30 0 0 36 0 7 30 0 0 0 36
,
 34 0 0 0 0 22 0 0 0 0 26 11 0 10 0 0 0 25 0 0 0 0 0 25 0 0 0 10 0 3 0 0 10 0 0 3

`G:=sub<GL(6,GF(37))| [1,0,0,9,7,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,36,0],[1,1,0,9,7,7,35,36,28,28,30,30,0,0,0,36,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[34,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,10,0,0,26,0,0,0,0,22,11,25,25,3,3] >;`

C422C18 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_2C_{18}`
`% in TeX`

`G:=Group("C4^2:2C18");`
`// GroupNames label`

`G:=SmallGroup(288,75);`
`// by ID`

`G=gap.SmallGroup(288,75);`
`# by ID`

`G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,50,6555,514,360,3784,3476,102,3036,5305]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^4=c^18=1,a*b=b*a,c*a*c^-1=a^2*b^-1,c*b*c^-1=a^-1*b^-1>;`
`// generators/relations`

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