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## G = C9⋊D4order 72 = 23·32

### The semidirect product of C9 and D4 acting via D4/C22=C2

Aliases: C92D4, Dic9⋊C2, D182C2, C6.10D6, C2.5D18, C222D9, C18.5C22, (C2×C18)⋊2C2, (C2×C6).3S3, C3.(C3⋊D4), SmallGroup(72,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C9⋊D4
 Chief series C1 — C3 — C9 — C18 — D18 — C9⋊D4
 Lower central C9 — C18 — C9⋊D4
 Upper central C1 — C2 — C22

Generators and relations for C9⋊D4
G = < a,b,c | a9=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Character table of C9⋊D4

 class 1 2A 2B 2C 3 4 6A 6B 6C 9A 9B 9C 18A 18B 18C 18D 18E 18F 18G 18H 18I size 1 1 2 18 2 18 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 ρ4 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 ρ5 2 2 2 0 2 0 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 -2 0 0 2 0 0 -2 0 2 2 2 -2 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ7 2 2 -2 0 2 0 -2 2 -2 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ8 2 2 -2 0 -1 0 1 -1 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D18 ρ9 2 2 -2 0 -1 0 1 -1 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D18 ρ10 2 2 2 0 -1 0 -1 -1 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ11 2 2 2 0 -1 0 -1 -1 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ12 2 2 -2 0 -1 0 1 -1 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D18 ρ13 2 2 2 0 -1 0 -1 -1 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ14 2 -2 0 0 2 0 0 -2 0 -1 -1 -1 1 -√-3 -√-3 -√-3 √-3 √-3 √-3 1 1 complex lifted from C3⋊D4 ρ15 2 -2 0 0 -1 0 -√-3 1 √-3 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ95-ζ94 ζ98-ζ9 -ζ97+ζ92 ζ95-ζ94 ζ97-ζ92 -ζ98+ζ9 -ζ95+ζ94 -ζ98-ζ9 -ζ97-ζ92 complex faithful ρ16 2 -2 0 0 2 0 0 -2 0 -1 -1 -1 1 √-3 √-3 √-3 -√-3 -√-3 -√-3 1 1 complex lifted from C3⋊D4 ρ17 2 -2 0 0 -1 0 -√-3 1 √-3 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ98-ζ9 -ζ97+ζ92 ζ95-ζ94 ζ98-ζ9 -ζ95+ζ94 ζ97-ζ92 -ζ98+ζ9 -ζ97-ζ92 -ζ95-ζ94 complex faithful ρ18 2 -2 0 0 -1 0 -√-3 1 √-3 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ97-ζ92 ζ95-ζ94 ζ98-ζ9 -ζ97+ζ92 -ζ98+ζ9 -ζ95+ζ94 ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 complex faithful ρ19 2 -2 0 0 -1 0 √-3 1 -√-3 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ95-ζ94 -ζ98+ζ9 ζ97-ζ92 -ζ95+ζ94 -ζ97+ζ92 ζ98-ζ9 ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 complex faithful ρ20 2 -2 0 0 -1 0 √-3 1 -√-3 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ98-ζ9 ζ97-ζ92 -ζ95+ζ94 -ζ98+ζ9 ζ95-ζ94 -ζ97+ζ92 ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 complex faithful ρ21 2 -2 0 0 -1 0 √-3 1 -√-3 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ97-ζ92 -ζ95+ζ94 -ζ98+ζ9 ζ97-ζ92 ζ98-ζ9 ζ95-ζ94 -ζ97+ζ92 -ζ95-ζ94 -ζ98-ζ9 complex faithful

Smallest permutation representation of C9⋊D4
On 36 points
Generators in S36
```(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)
(1 20 11 29)(2 19 12 28)(3 27 13 36)(4 26 14 35)(5 25 15 34)(6 24 16 33)(7 23 17 32)(8 22 18 31)(9 21 10 30)
(2 9)(3 8)(4 7)(5 6)(10 12)(13 18)(14 17)(15 16)(19 30)(20 29)(21 28)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)```

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

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

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

C9⋊D4 is a maximal subgroup of
D365C2  D4×D9  D42D9  C27⋊D4  C9.S4  D6⋊D9  C9⋊D12  Dic9⋊C6  C6.D18  C9⋊S4  Q8.D18  C23.D18  C45⋊D4  C9⋊D20  C457D4
C9⋊D4 is a maximal quotient of
Dic9⋊C4  D18⋊C4  D4.D9  D4⋊D9  C9⋊Q16  Q82D9  C18.D4  C27⋊D4  D6⋊D9  C9⋊D12  C6.D18  C23.D18  C45⋊D4  C9⋊D20  C457D4

Matrix representation of C9⋊D4 in GL2(𝔽19) generated by

 8 15 15 14
,
 12 4 16 7
,
 18 11 0 1
`G:=sub<GL(2,GF(19))| [8,15,15,14],[12,16,4,7],[18,0,11,1] >;`

C9⋊D4 in GAP, Magma, Sage, TeX

`C_9\rtimes D_4`
`% in TeX`

`G:=Group("C9:D4");`
`// GroupNames label`

`G:=SmallGroup(72,8);`
`// by ID`

`G=gap.SmallGroup(72,8);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,-3,61,803,138,1204]);`
`// Polycyclic`

`G:=Group<a,b,c|a^9=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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