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## G = C2.(C42⋊C9)  order 288 = 25·32

### The central stem extension by C2 of C42⋊C9

Aliases: C2.(C42⋊C9), C2.C42⋊C9, C22.(Q8⋊C9), C6.1(C42⋊C3), (C22×C6).7A4, C3.(C23.3A4), C23.3(C3.A4), (C2×C6).1SL2(𝔽3), (C3×C2.C42).C3, SmallGroup(288,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2.C42 — C2.(C42⋊C9)
 Chief series C1 — C2 — C23 — C2.C42 — C3×C2.C42 — C2.(C42⋊C9)
 Lower central C2.C42 — C2.(C42⋊C9)
 Upper central C1 — C6

Generators and relations for C2.(C42⋊C9)
G = < a,b,c,d | a2=b4=c4=d9=1, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=bc-1, dcd-1=b-1c2 >

Smallest permutation representation of C2.(C42⋊C9)
On 36 points
Generators in S36
```(1 25)(2 26)(3 27)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 36)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)
(2 33 26 16)(3 17)(5 36 20 10)(6 11)(8 30 23 13)(9 14)(12 29)(15 32)(18 35)(21 28)(24 31)(27 34)
(1 32)(3 17 27 34)(4 35)(6 11 21 28)(7 29)(9 14 24 31)(10 36)(12 22)(13 30)(15 25)(16 33)(18 19)
(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)| (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,36)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35), (2,33,26,16)(3,17)(5,36,20,10)(6,11)(8,30,23,13)(9,14)(12,29)(15,32)(18,35)(21,28)(24,31)(27,34), (1,32)(3,17,27,34)(4,35)(6,11,21,28)(7,29)(9,14,24,31)(10,36)(12,22)(13,30)(15,25)(16,33)(18,19), (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( (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,36)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35), (2,33,26,16)(3,17)(5,36,20,10)(6,11)(8,30,23,13)(9,14)(12,29)(15,32)(18,35)(21,28)(24,31)(27,34), (1,32)(3,17,27,34)(4,35)(6,11,21,28)(7,29)(9,14,24,31)(10,36)(12,22)(13,30)(15,25)(16,33)(18,19), (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([[(1,25),(2,26),(3,27),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,36),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35)], [(2,33,26,16),(3,17),(5,36,20,10),(6,11),(8,30,23,13),(9,14),(12,29),(15,32),(18,35),(21,28),(24,31),(27,34)], [(1,32),(3,17,27,34),(4,35),(6,11,21,28),(7,29),(9,14,24,31),(10,36),(12,22),(13,30),(15,25),(16,33),(18,19)], [(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)]])`

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 9A ··· 9F 12A ··· 12H 18A ··· 18F order 1 2 2 2 3 3 4 4 4 4 6 6 6 6 6 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 3 3 1 1 6 6 6 6 1 1 3 3 3 3 16 ··· 16 6 ··· 6 16 ··· 16

36 irreducible representations

 dim 1 1 1 2 2 2 3 3 3 3 6 6 type + - + + image C1 C3 C9 SL2(𝔽3) SL2(𝔽3) Q8⋊C9 A4 C3.A4 C42⋊C3 C42⋊C9 C23.3A4 C2.(C42⋊C9) kernel C2.(C42⋊C9) C3×C2.C42 C2.C42 C2×C6 C2×C6 C22 C22×C6 C23 C6 C2 C3 C1 # reps 1 2 6 1 2 6 1 2 4 8 1 2

Matrix representation of C2.(C42⋊C9) in GL5(𝔽37)

 36 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 11 0 0 0 10 0 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 36
,
 27 10 0 0 0 1 10 0 0 0 0 0 31 0 0 0 0 0 1 0 0 0 0 0 6
,
 28 21 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 15 0 0 7 0 0

`G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,10,0,0,0,11,0,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,36],[27,1,0,0,0,10,10,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,6],[28,12,0,0,0,21,0,0,0,0,0,0,0,0,7,0,0,8,0,0,0,0,0,15,0] >;`

C2.(C42⋊C9) in GAP, Magma, Sage, TeX

`C_2.(C_4^2\rtimes C_9)`
`% in TeX`

`G:=Group("C2.(C4^2:C9)");`
`// GroupNames label`

`G:=SmallGroup(288,3);`
`// by ID`

`G=gap.SmallGroup(288,3);`
`# by ID`

`G:=PCGroup([7,-3,-3,-2,2,-2,2,-2,21,380,268,2775,521,80,7564,10589]);`
`// Polycyclic`

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

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