Copied to
clipboard

## G = C32⋊3M5(2)  order 288 = 25·32

### The semidirect product of C32 and M5(2) acting via M5(2)/C8=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C32⋊3M5(2)
 Chief series C1 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊2C16 — C32⋊3M5(2)
 Lower central C32 — C3×C6 — C32⋊3M5(2)
 Upper central C1 — C4 — C8

Generators and relations for C323M5(2)
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=ab-1, dad=a-1, cbc-1=a-1b-1, dbd=b-1, dcd=c9 >

Smallest permutation representation of C323M5(2)
On 48 points
Generators in S48
```(1 24 36)(2 25 37)(3 38 26)(4 39 27)(5 28 40)(6 29 41)(7 42 30)(8 43 31)(9 32 44)(10 17 45)(11 46 18)(12 47 19)(13 20 48)(14 21 33)(15 34 22)(16 35 23)
(2 37 25)(4 27 39)(6 41 29)(8 31 43)(10 45 17)(12 19 47)(14 33 21)(16 23 35)
(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)
(1 9)(3 11)(5 13)(7 15)(17 45)(18 38)(19 47)(20 40)(21 33)(22 42)(23 35)(24 44)(25 37)(26 46)(27 39)(28 48)(29 41)(30 34)(31 43)(32 36)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 8A 8B 8C 8D 8E 8F 12A 12B 12C 12D 16A ··· 16H 24A ··· 24H order 1 2 2 3 3 4 4 4 6 6 8 8 8 8 8 8 12 12 12 12 16 ··· 16 24 ··· 24 size 1 1 18 4 4 1 1 18 4 4 2 2 9 9 9 9 4 4 4 4 18 ··· 18 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 4 4 4 type + + + + + image C1 C2 C2 C4 C4 C8 C8 M5(2) C32⋊C4 C2×C32⋊C4 C3⋊S3⋊3C8 C32⋊3M5(2) kernel C32⋊3M5(2) C32⋊2C16 C8×C3⋊S3 C3×C24 C4×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C32 C8 C4 C2 C1 # reps 1 2 1 2 2 4 4 4 2 2 4 8

Matrix representation of C323M5(2) in GL6(𝔽97)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 96 1 0 0 0 0 96 0 0 0 0 0 0 0 96 1 0 0 0 0 96 0
,
 1 0 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 0 96 0 0 0 0 1 96
,
 0 96 0 0 0 0 50 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 22 0 0 0 0 0 0 22 0 0
,
 96 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96],[0,50,0,0,0,0,96,0,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,1,0,0,0,0,1,0,0,0],[96,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C323M5(2) in GAP, Magma, Sage, TeX

`C_3^2\rtimes_3M_5(2)`
`% in TeX`

`G:=Group("C3^2:3M5(2)");`
`// GroupNames label`

`G:=SmallGroup(288,413);`
`// by ID`

`G=gap.SmallGroup(288,413);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,64,58,80,9413,691,12550,2372]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=a^-1,c*b*c^-1=a^-1*b^-1,d*b*d=b^-1,d*c*d=c^9>;`
`// generators/relations`

Export

׿
×
𝔽