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## G = C42⋊C18order 288 = 25·32

### 1st semidirect product of C42 and C18 acting via C18/C3=C6

Aliases: C421C18, C42⋊C91C2, C422C2⋊C9, (C4×C12).1C6, (C22×C6).2A4, C3.(C42⋊C6), C23.1(C3.A4), (C2×C6).7(C2×A4), (C3×C422C2).C3, C22.3(C2×C3.A4), SmallGroup(288,74)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C42⋊C18
G = < a,b,c | a4=b4=c18=1, ab=ba, cac-1=b-1, cbc-1=a-1b >

Character table of C42⋊C18

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 6C 6D 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 18A 18B 18C 18D 18E 18F size 1 3 4 1 1 6 6 12 3 3 4 4 16 16 16 16 16 16 6 6 6 6 12 12 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 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 1 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 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 1 1 1 1 -1 -1 ζ65 ζ6 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ6 1 1 -1 1 1 1 1 -1 1 1 -1 -1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 1 1 1 1 -1 -1 ζ6 ζ65 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ7 1 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ98 ζ95 ζ97 ζ9 ζ94 ζ92 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ95 ζ97 ζ9 ζ98 ζ92 ζ94 linear of order 9 ρ8 1 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ92 ζ98 ζ94 ζ97 ζ9 ζ95 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ98 ζ94 ζ97 ζ92 ζ95 ζ9 linear of order 9 ρ9 1 1 -1 ζ32 ζ3 1 1 -1 ζ32 ζ3 ζ65 ζ6 ζ94 ζ97 ζ98 ζ95 ζ92 ζ9 ζ3 ζ3 ζ32 ζ32 ζ65 ζ6 -ζ97 -ζ98 -ζ95 -ζ94 -ζ9 -ζ92 linear of order 18 ρ10 1 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ94 ζ97 ζ98 ζ95 ζ92 ζ9 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ97 ζ98 ζ95 ζ94 ζ9 ζ92 linear of order 9 ρ11 1 1 -1 ζ3 ζ32 1 1 -1 ζ3 ζ32 ζ6 ζ65 ζ95 ζ92 ζ9 ζ94 ζ97 ζ98 ζ32 ζ32 ζ3 ζ3 ζ6 ζ65 -ζ92 -ζ9 -ζ94 -ζ95 -ζ98 -ζ97 linear of order 18 ρ12 1 1 -1 ζ32 ζ3 1 1 -1 ζ32 ζ3 ζ65 ζ6 ζ9 ζ94 ζ92 ζ98 ζ95 ζ97 ζ3 ζ3 ζ32 ζ32 ζ65 ζ6 -ζ94 -ζ92 -ζ98 -ζ9 -ζ97 -ζ95 linear of order 18 ρ13 1 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ9 ζ94 ζ92 ζ98 ζ95 ζ97 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ94 ζ92 ζ98 ζ9 ζ97 ζ95 linear of order 9 ρ14 1 1 -1 ζ3 ζ32 1 1 -1 ζ3 ζ32 ζ6 ζ65 ζ92 ζ98 ζ94 ζ97 ζ9 ζ95 ζ32 ζ32 ζ3 ζ3 ζ6 ζ65 -ζ98 -ζ94 -ζ97 -ζ92 -ζ95 -ζ9 linear of order 18 ρ15 1 1 -1 ζ3 ζ32 1 1 -1 ζ3 ζ32 ζ6 ζ65 ζ98 ζ95 ζ97 ζ9 ζ94 ζ92 ζ32 ζ32 ζ3 ζ3 ζ6 ζ65 -ζ95 -ζ97 -ζ9 -ζ98 -ζ92 -ζ94 linear of order 18 ρ16 1 1 -1 ζ32 ζ3 1 1 -1 ζ32 ζ3 ζ65 ζ6 ζ97 ζ9 ζ95 ζ92 ζ98 ζ94 ζ3 ζ3 ζ32 ζ32 ζ65 ζ6 -ζ9 -ζ95 -ζ92 -ζ97 -ζ94 -ζ98 linear of order 18 ρ17 1 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ95 ζ92 ζ9 ζ94 ζ97 ζ98 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ92 ζ9 ζ94 ζ95 ζ98 ζ97 linear of order 9 ρ18 1 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ97 ζ9 ζ95 ζ92 ζ98 ζ94 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ9 ζ95 ζ92 ζ97 ζ94 ζ98 linear of order 9 ρ19 3 3 -3 3 3 -1 -1 1 3 3 -3 -3 0 0 0 0 0 0 -1 -1 -1 -1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ20 3 3 3 3 3 -1 -1 -1 3 3 3 3 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ21 3 3 3 -3+3√-3/2 -3-3√-3/2 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ6 ζ6 ζ65 ζ65 ζ6 ζ65 0 0 0 0 0 0 complex lifted from C3.A4 ρ22 3 3 -3 -3-3√-3/2 -3+3√-3/2 -1 -1 1 -3-3√-3/2 -3+3√-3/2 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 ζ65 ζ65 ζ6 ζ6 ζ3 ζ32 0 0 0 0 0 0 complex lifted from C2×C3.A4 ρ23 3 3 3 -3-3√-3/2 -3+3√-3/2 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ65 ζ65 ζ6 ζ6 ζ65 ζ6 0 0 0 0 0 0 complex lifted from C3.A4 ρ24 3 3 -3 -3+3√-3/2 -3-3√-3/2 -1 -1 1 -3+3√-3/2 -3-3√-3/2 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 ζ6 ζ6 ζ65 ζ65 ζ32 ζ3 0 0 0 0 0 0 complex lifted from C2×C3.A4 ρ25 6 -2 0 6 6 2i -2i 0 -2 -2 0 0 0 0 0 0 0 0 -2i 2i -2i 2i 0 0 0 0 0 0 0 0 complex lifted from C42⋊C6 ρ26 6 -2 0 6 6 -2i 2i 0 -2 -2 0 0 0 0 0 0 0 0 2i -2i 2i -2i 0 0 0 0 0 0 0 0 complex lifted from C42⋊C6 ρ27 6 -2 0 -3+3√-3 -3-3√-3 -2i 2i 0 1-√-3 1+√-3 0 0 0 0 0 0 0 0 2ζ4ζ32 2ζ43ζ32 2ζ4ζ3 2ζ43ζ3 0 0 0 0 0 0 0 0 complex faithful ρ28 6 -2 0 -3-3√-3 -3+3√-3 2i -2i 0 1+√-3 1-√-3 0 0 0 0 0 0 0 0 2ζ43ζ3 2ζ4ζ3 2ζ43ζ32 2ζ4ζ32 0 0 0 0 0 0 0 0 complex faithful ρ29 6 -2 0 -3+3√-3 -3-3√-3 2i -2i 0 1-√-3 1+√-3 0 0 0 0 0 0 0 0 2ζ43ζ32 2ζ4ζ32 2ζ43ζ3 2ζ4ζ3 0 0 0 0 0 0 0 0 complex faithful ρ30 6 -2 0 -3-3√-3 -3+3√-3 -2i 2i 0 1+√-3 1-√-3 0 0 0 0 0 0 0 0 2ζ4ζ3 2ζ43ζ3 2ζ4ζ32 2ζ43ζ32 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C42⋊C18
On 72 points
Generators in S72
```(1 33 24 10)(2 61 25 42)(3 52)(4 13 27 36)(5 64 28 45)(6 56)(7 21 30 16)(8 67 31 48)(9 40)(11 70 34 51)(12 62)(14 55 19 54)(15 46)(17 58 22 39)(18 68)(20 65)(23 49)(26 71)(29 37)(32 59)(35 43)(38 47 57 66)(41 69 60 50)(44 53 63 72)
(1 41 24 60)(2 51)(3 35 26 12)(4 44 27 63)(5 55)(6 15 29 20)(7 47 30 66)(8 39)(9 23 32 18)(10 50 33 69)(11 61)(13 53 36 72)(14 45)(16 38 21 57)(17 67)(19 64)(22 48)(25 70)(28 54)(31 58)(34 42)(37 65 56 46)(40 49 59 68)(43 71 62 52)
(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)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)```

`G:=sub<Sym(72)| (1,33,24,10)(2,61,25,42)(3,52)(4,13,27,36)(5,64,28,45)(6,56)(7,21,30,16)(8,67,31,48)(9,40)(11,70,34,51)(12,62)(14,55,19,54)(15,46)(17,58,22,39)(18,68)(20,65)(23,49)(26,71)(29,37)(32,59)(35,43)(38,47,57,66)(41,69,60,50)(44,53,63,72), (1,41,24,60)(2,51)(3,35,26,12)(4,44,27,63)(5,55)(6,15,29,20)(7,47,30,66)(8,39)(9,23,32,18)(10,50,33,69)(11,61)(13,53,36,72)(14,45)(16,38,21,57)(17,67)(19,64)(22,48)(25,70)(28,54)(31,58)(34,42)(37,65,56,46)(40,49,59,68)(43,71,62,52), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)>;`

`G:=Group( (1,33,24,10)(2,61,25,42)(3,52)(4,13,27,36)(5,64,28,45)(6,56)(7,21,30,16)(8,67,31,48)(9,40)(11,70,34,51)(12,62)(14,55,19,54)(15,46)(17,58,22,39)(18,68)(20,65)(23,49)(26,71)(29,37)(32,59)(35,43)(38,47,57,66)(41,69,60,50)(44,53,63,72), (1,41,24,60)(2,51)(3,35,26,12)(4,44,27,63)(5,55)(6,15,29,20)(7,47,30,66)(8,39)(9,23,32,18)(10,50,33,69)(11,61)(13,53,36,72)(14,45)(16,38,21,57)(17,67)(19,64)(22,48)(25,70)(28,54)(31,58)(34,42)(37,65,56,46)(40,49,59,68)(43,71,62,52), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72) );`

`G=PermutationGroup([[(1,33,24,10),(2,61,25,42),(3,52),(4,13,27,36),(5,64,28,45),(6,56),(7,21,30,16),(8,67,31,48),(9,40),(11,70,34,51),(12,62),(14,55,19,54),(15,46),(17,58,22,39),(18,68),(20,65),(23,49),(26,71),(29,37),(32,59),(35,43),(38,47,57,66),(41,69,60,50),(44,53,63,72)], [(1,41,24,60),(2,51),(3,35,26,12),(4,44,27,63),(5,55),(6,15,29,20),(7,47,30,66),(8,39),(9,23,32,18),(10,50,33,69),(11,61),(13,53,36,72),(14,45),(16,38,21,57),(17,67),(19,64),(22,48),(25,70),(28,54),(31,58),(34,42),(37,65,56,46),(40,49,59,68),(43,71,62,52)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)]])`

Matrix representation of C42⋊C18 in GL6(𝔽37)

 1 35 0 0 0 0 1 36 0 0 0 0 9 11 6 0 0 0 9 11 0 6 0 0 12 27 0 0 0 6 35 27 0 0 31 0
,
 31 0 0 0 0 0 0 31 0 0 0 0 0 0 0 31 0 0 29 0 6 0 0 0 2 0 0 0 0 1 12 0 0 0 36 0
,
 1 0 0 0 0 22 0 0 0 0 26 11 8 10 0 0 0 27 8 0 0 0 0 27 23 0 0 10 0 36 23 0 10 0 0 36

`G:=sub<GL(6,GF(37))| [1,1,9,9,12,35,35,36,11,11,27,27,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,31,0,0,0,0,6,0],[31,0,0,29,2,12,0,31,0,0,0,0,0,0,0,6,0,0,0,0,31,0,0,0,0,0,0,0,0,36,0,0,0,0,1,0],[1,0,8,8,23,23,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,10,0,0,26,0,0,0,0,22,11,27,27,36,36] >;`

C42⋊C18 in GAP, Magma, Sage, TeX

`C_4^2\rtimes C_{18}`
`% in TeX`

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

`G:=SmallGroup(288,74);`
`// by ID`

`G=gap.SmallGroup(288,74);`
`# by ID`

`G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,50,2523,514,360,6304,3476,102,3036,5305]);`
`// Polycyclic`

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

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