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## G = D9⋊A4order 216 = 23·33

### The semidirect product of D9 and A4 acting via A4/C22=C3

Aliases: D9⋊A4, C9⋊A4⋊C2, C9⋊(C2×A4), (C3×A4).S3, (C2×C18)⋊1C6, C3.1(S3×A4), C222(C9⋊C6), (C22×D9)⋊1C3, (C2×C6).5(C3×S3), SmallGroup(216,96)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — D9⋊A4
 Chief series C1 — C3 — C9 — C2×C18 — C9⋊A4 — D9⋊A4
 Lower central C2×C18 — D9⋊A4
 Upper central C1

Generators and relations for D9⋊A4
G = < a,b,c,d,e | a9=b2=c2=d2=e3=1, bab=a-1, ac=ca, ad=da, eae-1=a4, bc=cb, bd=db, ebe-1=a3b, ece-1=cd=dc, ede-1=c >

3C2
9C2
27C2
12C3
27C22
27C22
3S3
3C6
9S3
36C6
4C32
8C9
9C23
3A4
9D6
9D6
3D9
3C18
12C3×S3
3D18
3D18

Character table of D9⋊A4

 class 1 2A 2B 2C 3A 3B 3C 6A 6B 6C 9A 9B 9C 18A 18B 18C size 1 3 9 27 2 12 12 6 36 36 6 24 24 6 6 6 ρ1 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 linear of order 2 ρ3 1 1 1 1 1 ζ3 ζ32 1 ζ32 ζ3 1 ζ32 ζ3 1 1 1 linear of order 3 ρ4 1 1 1 1 1 ζ32 ζ3 1 ζ3 ζ32 1 ζ3 ζ32 1 1 1 linear of order 3 ρ5 1 1 -1 -1 1 ζ32 ζ3 1 ζ65 ζ6 1 ζ3 ζ32 1 1 1 linear of order 6 ρ6 1 1 -1 -1 1 ζ3 ζ32 1 ζ6 ζ65 1 ζ32 ζ3 1 1 1 linear of order 6 ρ7 2 2 0 0 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 0 2 -1+√-3 -1-√-3 2 0 0 -1 ζ6 ζ65 -1 -1 -1 complex lifted from C3×S3 ρ9 2 2 0 0 2 -1-√-3 -1+√-3 2 0 0 -1 ζ65 ζ6 -1 -1 -1 complex lifted from C3×S3 ρ10 3 -1 -3 1 3 0 0 -1 0 0 3 0 0 -1 -1 -1 orthogonal lifted from C2×A4 ρ11 3 -1 3 -1 3 0 0 -1 0 0 3 0 0 -1 -1 -1 orthogonal lifted from A4 ρ12 6 6 0 0 -3 0 0 -3 0 0 0 0 0 0 0 0 orthogonal lifted from C9⋊C6 ρ13 6 -2 0 0 6 0 0 -2 0 0 -3 0 0 1 1 1 orthogonal lifted from S3×A4 ρ14 6 -2 0 0 -3 0 0 1 0 0 0 0 0 2ζ97+2ζ92 2ζ95+2ζ94 2ζ98+2ζ9 orthogonal faithful ρ15 6 -2 0 0 -3 0 0 1 0 0 0 0 0 2ζ98+2ζ9 2ζ97+2ζ92 2ζ95+2ζ94 orthogonal faithful ρ16 6 -2 0 0 -3 0 0 1 0 0 0 0 0 2ζ95+2ζ94 2ζ98+2ζ9 2ζ97+2ζ92 orthogonal faithful

Smallest permutation representation of D9⋊A4
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 9)(2 8)(3 7)(4 6)(10 14)(11 13)(15 18)(16 17)(19 23)(20 22)(24 27)(25 26)(28 32)(29 31)(33 36)(34 35)
(1 17)(2 18)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(1 26)(2 27)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(2 8 5)(3 6 9)(10 22 34)(11 20 29)(12 27 33)(13 25 28)(14 23 32)(15 21 36)(16 19 31)(17 26 35)(18 24 30)```

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

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

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

D9⋊A4 is a maximal quotient of   Dic9.A4  D18.A4  Dic9⋊A4

Matrix representation of D9⋊A4 in GL6(𝔽19)

 0 0 18 18 0 0 0 0 1 0 0 0 0 0 0 0 18 18 0 0 0 0 1 0 0 1 0 0 0 0 18 18 0 0 0 0
,
 0 0 18 18 0 0 0 0 0 1 0 0 18 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 18 18
,
 6 0 3 14 14 3 0 6 5 8 16 11 8 5 6 0 3 14 14 3 0 6 5 8 11 16 8 5 6 0 3 14 14 3 0 6
,
 6 0 5 8 8 5 0 6 11 16 14 3 16 11 6 0 5 8 8 5 0 6 11 16 3 14 16 11 6 0 5 8 8 5 0 6
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 18 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 18 18

`G:=sub<GL(6,GF(19))| [0,0,0,0,0,18,0,0,0,0,1,18,18,1,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0],[0,0,18,0,0,0,0,0,18,1,0,0,18,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,18],[6,0,8,14,11,3,0,6,5,3,16,14,3,5,6,0,8,14,14,8,0,6,5,3,14,16,3,5,6,0,3,11,14,8,0,6],[6,0,16,8,3,5,0,6,11,5,14,8,5,11,6,0,16,8,8,16,0,6,11,5,8,14,5,11,6,0,5,3,8,16,0,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,1,18] >;`

D9⋊A4 in GAP, Magma, Sage, TeX

`D_9\rtimes A_4`
`% in TeX`

`G:=Group("D9:A4");`
`// GroupNames label`

`G:=SmallGroup(216,96);`
`// by ID`

`G=gap.SmallGroup(216,96);`
`# by ID`

`G:=PCGroup([6,-2,-3,-2,2,-3,-3,170,81,3604,1450,208,5189]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^9=b^2=c^2=d^2=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^4,b*c=c*b,b*d=d*b,e*b*e^-1=a^3*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;`
`// generators/relations`

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