Copied to
clipboard

## G = C4.3F9order 288 = 25·32

### 2nd central extension by C4 of F9

Aliases: C4.3F9, C3⋊S3⋊C16, C2.F93C2, C2.1(C2×F9), (C3×C12).2C8, C321(C2×C16), C322C8.1C4, C322C8.3C22, (C4×C3⋊S3).1C4, (C2×C3⋊S3).1C8, (C3×C6).1(C2×C8), C3⋊S33C8.8C2, C3⋊Dic3.1(C2×C4), SmallGroup(288,861)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4.3F9
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2.F9 — C4.3F9
 Lower central C32 — C4.3F9
 Upper central C1 — C4

Generators and relations for C4.3F9
G = < a,b,c,d | a4=b3=c3=1, d8=a2, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of C4.3F9
On 48 points
Generators in S48
```(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 34 25 42)(18 35 26 43)(19 36 27 44)(20 37 28 45)(21 38 29 46)(22 39 30 47)(23 40 31 48)(24 41 32 33)
(2 21 42)(3 22 43)(4 44 23)(6 46 25)(7 47 26)(8 27 48)(10 29 34)(11 30 35)(12 36 31)(14 38 17)(15 39 18)(16 19 40)
(1 20 41)(3 22 43)(4 23 44)(5 45 24)(7 47 26)(8 48 27)(9 28 33)(11 30 35)(12 31 36)(13 37 32)(15 39 18)(16 40 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 37 38 39 40 41 42 43 44 45 46 47 48)```

`G:=sub<Sym(48)| (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,34,25,42)(18,35,26,43)(19,36,27,44)(20,37,28,45)(21,38,29,46)(22,39,30,47)(23,40,31,48)(24,41,32,33), (2,21,42)(3,22,43)(4,44,23)(6,46,25)(7,47,26)(8,27,48)(10,29,34)(11,30,35)(12,36,31)(14,38,17)(15,39,18)(16,19,40), (1,20,41)(3,22,43)(4,23,44)(5,45,24)(7,47,26)(8,48,27)(9,28,33)(11,30,35)(12,31,36)(13,37,32)(15,39,18)(16,40,19), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)>;`

`G:=Group( (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,34,25,42)(18,35,26,43)(19,36,27,44)(20,37,28,45)(21,38,29,46)(22,39,30,47)(23,40,31,48)(24,41,32,33), (2,21,42)(3,22,43)(4,44,23)(6,46,25)(7,47,26)(8,27,48)(10,29,34)(11,30,35)(12,36,31)(14,38,17)(15,39,18)(16,19,40), (1,20,41)(3,22,43)(4,23,44)(5,45,24)(7,47,26)(8,48,27)(9,28,33)(11,30,35)(12,31,36)(13,37,32)(15,39,18)(16,40,19), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48) );`

`G=PermutationGroup([(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8),(17,34,25,42),(18,35,26,43),(19,36,27,44),(20,37,28,45),(21,38,29,46),(22,39,30,47),(23,40,31,48),(24,41,32,33)], [(2,21,42),(3,22,43),(4,44,23),(6,46,25),(7,47,26),(8,27,48),(10,29,34),(11,30,35),(12,36,31),(14,38,17),(15,39,18),(16,19,40)], [(1,20,41),(3,22,43),(4,23,44),(5,45,24),(7,47,26),(8,48,27),(9,28,33),(11,30,35),(12,31,36),(13,37,32),(15,39,18),(16,40,19)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)])`

36 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 8A ··· 8H 12A 12B 16A ··· 16P order 1 2 2 2 3 4 4 4 4 6 8 ··· 8 12 12 16 ··· 16 size 1 1 9 9 8 1 1 9 9 8 9 ··· 9 8 8 9 ··· 9

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 8 8 8 type + + + + + image C1 C2 C2 C4 C4 C8 C8 C16 F9 C2×F9 C4.3F9 kernel C4.3F9 C2.F9 C3⋊S3⋊3C8 C32⋊2C8 C4×C3⋊S3 C3×C12 C2×C3⋊S3 C3⋊S3 C4 C2 C1 # reps 1 2 1 2 2 4 4 16 1 1 2

Matrix representation of C4.3F9 in GL9(𝔽97)

 75 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 96 96 0 0 0 0 0 1 1 0 0 96 96 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 96 96
,
 1 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 1 96 0 0 0 0 0 0 0 0 96 0 1 0 0 0 0 0 1 0 96 96 0 0 0 0 0 0 96 0 0 0 1 0 0 0 1 0 0 0 96 96 0 0 0 0 96 0 0 0 0 1 0 0 0 96 0 0 0 0 0 1
,
 70 0 0 0 0 0 0 0 0 0 0 0 0 0 96 1 0 0 0 1 1 0 0 95 96 0 0 0 0 0 0 0 96 0 1 0 0 0 0 0 0 96 0 0 1 0 0 0 1 0 96 0 0 0 0 0 0 0 1 96 0 0 0 0 0 0 0 0 96 0 0 0 0 0 1 0 0 96 0 0 0

`G:=sub<GL(9,GF(97))| [75,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,96,1,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,96,96,96,0,96,0,96,96,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[70,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,96,95,96,96,96,96,96,96,0,1,96,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C4.3F9 in GAP, Magma, Sage, TeX

`C_4._3F_9`
`% in TeX`

`G:=Group("C4.3F9");`
`// GroupNames label`

`G:=SmallGroup(288,861);`
`// by ID`

`G=gap.SmallGroup(288,861);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,120,58,80,4037,2371,362,10982,3156,1203]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^3=c^3=1,d^8=a^2,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;`
`// generators/relations`

Export

׿
×
𝔽