Copied to
clipboard

## G = C92⋊4S3order 486 = 2·35

### 4th semidirect product of C92 and S3 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C32⋊C9 — C92⋊4S3
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C92⋊3C3 — C92⋊4S3
 Lower central C32⋊C9 — C92⋊4S3
 Upper central C1 — C9

Generators and relations for C924S3
G = < a,b,c,d | a9=b9=c3=d2=1, ab=ba, ac=ca, ad=da, cbc-1=a6b, dbd=b-1, dcd=c-1 >

Subgroups: 326 in 78 conjugacy classes, 24 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, C33, C3×D9, S3×C9, C3×C3⋊S3, C92, C32⋊C9, C32⋊C9, C9⋊C9, C32×C9, C9×D9, C322D9, C9×C3⋊S3, C923C3, C924S3
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C9⋊S3, C3×C3⋊S3, C3×C9⋊S3, He3.4C6, C924S3

Smallest permutation representation of C924S3
On 54 points
Generators in S54
```(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 49 50 51 52 53 54)
(1 44 34 4 38 28 7 41 31)(2 45 35 5 39 29 8 42 32)(3 37 36 6 40 30 9 43 33)(10 52 23 16 49 20 13 46 26)(11 53 24 17 50 21 14 47 27)(12 54 25 18 51 22 15 48 19)
(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 10)(8 11)(9 12)(19 43)(20 44)(21 45)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 52)(29 53)(30 54)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)```

`G:=sub<Sym(54)| (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)(46,47,48,49,50,51,52,53,54), (1,44,34,4,38,28,7,41,31)(2,45,35,5,39,29,8,42,32)(3,37,36,6,40,30,9,43,33)(10,52,23,16,49,20,13,46,26)(11,53,24,17,50,21,14,47,27)(12,54,25,18,51,22,15,48,19), (19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,10)(8,11)(9,12)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,52)(29,53)(30,54)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)>;`

`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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,44,34,4,38,28,7,41,31)(2,45,35,5,39,29,8,42,32)(3,37,36,6,40,30,9,43,33)(10,52,23,16,49,20,13,46,26)(11,53,24,17,50,21,14,47,27)(12,54,25,18,51,22,15,48,19), (19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,10)(8,11)(9,12)(19,43)(20,44)(21,45)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,52)(29,53)(30,54)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51) );`

`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),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(1,44,34,4,38,28,7,41,31),(2,45,35,5,39,29,8,42,32),(3,37,36,6,40,30,9,43,33),(10,52,23,16,49,20,13,46,26),(11,53,24,17,50,21,14,47,27),(12,54,25,18,51,22,15,48,19)], [(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,10),(8,11),(9,12),(19,43),(20,44),(21,45),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,52),(29,53),(30,54),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51)]])`

63 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 9A ··· 9F 9G ··· 9L 9M ··· 9AS 18A ··· 18F order 1 2 3 3 3 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 2 2 6 6 6 27 27 1 ··· 1 2 ··· 2 6 ··· 6 27 ··· 27

63 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 6 type + + + + + image C1 C2 C3 C6 S3 S3 D9 C3×S3 C3×S3 C3×D9 He3.4C6 C92⋊4S3 kernel C92⋊4S3 C92⋊3C3 C32⋊2D9 C32⋊C9 C92 C32×C9 C3×C9 C3×C9 C33 C32 C3 C1 # reps 1 1 2 2 3 1 9 6 2 18 12 6

Matrix representation of C924S3 in GL5(𝔽19)

 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 18 3 0 0 0 16 8 0 0 0 0 0 0 0 9 0 0 6 0 0 0 0 0 6 0
,
 16 8 0 0 0 11 2 0 0 0 0 0 7 0 0 0 0 0 11 0 0 0 0 0 1
,
 18 0 0 0 0 16 1 0 0 0 0 0 0 3 0 0 0 13 0 0 0 0 0 0 18

`G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[18,16,0,0,0,3,8,0,0,0,0,0,0,6,0,0,0,0,0,6,0,0,9,0,0],[16,11,0,0,0,8,2,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,1],[18,16,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,3,0,0,0,0,0,0,18] >;`

C924S3 in GAP, Magma, Sage, TeX

`C_9^2\rtimes_4S_3`
`% in TeX`

`G:=Group("C9^2:4S3");`
`// GroupNames label`

`G:=SmallGroup(486,140);`
`// by ID`

`G=gap.SmallGroup(486,140);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1190,338,867,735,3244]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^9=b^9=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^6*b,d*b*d=b^-1,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽