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## G = C24⋊C9order 144 = 24·32

### 2nd semidirect product of C24 and C9 acting via C9/C3=C3

Aliases: C242C9, (C2×C6).3A4, C22⋊(C3.A4), C3.(C22⋊A4), (C23×C6).2C3, SmallGroup(144,111)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24⋊C9
 Chief series C1 — C22 — C24 — C23×C6 — C24⋊C9
 Lower central C24 — C24⋊C9
 Upper central C1 — C3

Generators and relations for C24⋊C9
G = < a,b,c,d,e | a2=b2=c2=d2=e9=1, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, ebe-1=a, ece-1=cd=dc, ede-1=c >

Subgroups: 171 in 61 conjugacy classes, 15 normal (5 characteristic)
C1, C2, C3, C22, C22, C6, C23, C9, C2×C6, C2×C6, C24, C22×C6, C3.A4, C23×C6, C24⋊C9
Quotients: C1, C3, C9, A4, C3.A4, C22⋊A4, C24⋊C9

Character table of C24⋊C9

 class 1 2A 2B 2C 2D 2E 3A 3B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 9A 9B 9C 9D 9E 9F size 1 3 3 3 3 3 1 1 3 3 3 3 3 3 3 3 3 3 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 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ4 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ92 ζ98 ζ95 ζ94 ζ9 ζ97 linear of order 9 ρ5 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ94 ζ97 ζ9 ζ98 ζ92 ζ95 linear of order 9 ρ6 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ98 ζ95 ζ92 ζ97 ζ94 ζ9 linear of order 9 ρ7 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ95 ζ92 ζ98 ζ9 ζ97 ζ94 linear of order 9 ρ8 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ97 ζ9 ζ94 ζ95 ζ98 ζ92 linear of order 9 ρ9 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ9 ζ94 ζ97 ζ92 ζ95 ζ98 linear of order 9 ρ10 3 -1 -1 -1 3 -1 3 3 -1 -1 3 -1 -1 -1 -1 -1 3 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ11 3 -1 3 -1 -1 -1 3 3 3 -1 -1 -1 -1 -1 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ12 3 3 -1 -1 -1 -1 3 3 -1 -1 -1 3 -1 -1 -1 -1 -1 3 0 0 0 0 0 0 orthogonal lifted from A4 ρ13 3 -1 -1 3 -1 -1 3 3 -1 3 -1 -1 -1 -1 -1 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ14 3 -1 -1 -1 -1 3 3 3 -1 -1 -1 -1 3 3 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ15 3 3 -1 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 ζ6 ζ6 ζ6 -3-3√-3/2 ζ6 ζ65 ζ65 ζ65 ζ65 -3+3√-3/2 0 0 0 0 0 0 complex lifted from C3.A4 ρ16 3 -1 -1 3 -1 -1 -3+3√-3/2 -3-3√-3/2 ζ6 -3-3√-3/2 ζ6 ζ6 ζ6 ζ65 ζ65 -3+3√-3/2 ζ65 ζ65 0 0 0 0 0 0 complex lifted from C3.A4 ρ17 3 -1 3 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 ζ6 ζ6 ζ6 ζ6 ζ65 -3+3√-3/2 ζ65 ζ65 ζ65 0 0 0 0 0 0 complex lifted from C3.A4 ρ18 3 -1 -1 -1 3 -1 -3-3√-3/2 -3+3√-3/2 ζ65 ζ65 -3+3√-3/2 ζ65 ζ65 ζ6 ζ6 ζ6 -3-3√-3/2 ζ6 0 0 0 0 0 0 complex lifted from C3.A4 ρ19 3 -1 -1 -1 -1 3 -3-3√-3/2 -3+3√-3/2 ζ65 ζ65 ζ65 ζ65 -3+3√-3/2 -3-3√-3/2 ζ6 ζ6 ζ6 ζ6 0 0 0 0 0 0 complex lifted from C3.A4 ρ20 3 -1 -1 3 -1 -1 -3-3√-3/2 -3+3√-3/2 ζ65 -3+3√-3/2 ζ65 ζ65 ζ65 ζ6 ζ6 -3-3√-3/2 ζ6 ζ6 0 0 0 0 0 0 complex lifted from C3.A4 ρ21 3 -1 -1 -1 3 -1 -3+3√-3/2 -3-3√-3/2 ζ6 ζ6 -3-3√-3/2 ζ6 ζ6 ζ65 ζ65 ζ65 -3+3√-3/2 ζ65 0 0 0 0 0 0 complex lifted from C3.A4 ρ22 3 3 -1 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 ζ65 ζ65 ζ65 -3+3√-3/2 ζ65 ζ6 ζ6 ζ6 ζ6 -3-3√-3/2 0 0 0 0 0 0 complex lifted from C3.A4 ρ23 3 -1 3 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 ζ65 ζ65 ζ65 ζ65 ζ6 -3-3√-3/2 ζ6 ζ6 ζ6 0 0 0 0 0 0 complex lifted from C3.A4 ρ24 3 -1 -1 -1 -1 3 -3+3√-3/2 -3-3√-3/2 ζ6 ζ6 ζ6 ζ6 -3-3√-3/2 -3+3√-3/2 ζ65 ζ65 ζ65 ζ65 0 0 0 0 0 0 complex lifted from C3.A4

Smallest permutation representation of C24⋊C9
On 36 points
Generators in S36
```(2 24)(3 25)(5 27)(6 19)(8 21)(9 22)(10 30)(11 31)(13 33)(14 34)(16 36)(17 28)
(1 23)(3 25)(4 26)(6 19)(7 20)(9 22)(11 31)(12 32)(14 34)(15 35)(17 28)(18 29)
(1 12)(3 14)(4 15)(6 17)(7 18)(9 11)(19 28)(20 29)(22 31)(23 32)(25 34)(26 35)
(1 12)(2 13)(4 15)(5 16)(7 18)(8 10)(20 29)(21 30)(23 32)(24 33)(26 35)(27 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)```

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

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

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

C24⋊C9 is a maximal subgroup of
C24⋊C18  C24⋊D9  A4×C3.A4  C3.A42  C24⋊3- 1+2  C2423- 1+2  C9×C22⋊A4  C2443- 1+2  C62.A4
C24⋊C9 is a maximal quotient of
C22⋊(Q8⋊C9)  2+ 1+42C9  C24⋊C27

Matrix representation of C24⋊C9 in GL6(𝔽19)

 1 0 0 0 0 0 0 18 0 0 0 0 14 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 8 0 18
,
 18 0 0 0 0 0 0 18 0 0 0 0 5 9 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 11 18 1
,
 18 0 0 0 0 0 0 18 0 0 0 0 5 9 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 18 0 0 0 0 0 0 1 0 0 0 0 0 10 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 14 10 17 0 0 0 0 0 9 0 0 0 0 0 0 0 1 0 0 0 0 4 10 18 0 0 0 10 9 9

`G:=sub<GL(6,GF(19))| [1,0,14,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,8,0,0,0,0,18,0,0,0,0,0,0,18],[18,0,5,0,0,0,0,18,9,0,0,0,0,0,1,0,0,0,0,0,0,18,0,11,0,0,0,0,18,18,0,0,0,0,0,1],[18,0,5,0,0,0,0,18,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,0,0,0,0,0,0,1,10,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,14,0,0,0,0,1,10,0,0,0,0,0,17,9,0,0,0,0,0,0,0,4,10,0,0,0,1,10,9,0,0,0,0,18,9] >;`

C24⋊C9 in GAP, Magma, Sage, TeX

`C_2^4\rtimes C_9`
`% in TeX`

`G:=Group("C2^4:C9");`
`// GroupNames label`

`G:=SmallGroup(144,111);`
`// by ID`

`G=gap.SmallGroup(144,111);`
`# by ID`

`G:=PCGroup([6,-3,-3,-2,2,-2,2,18,326,651,2164,3893]);`
`// Polycyclic`

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

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