Copied to
clipboard

G = 3- 1+4⋊2C2order 486 = 2·35

2nd semidirect product of 3- 1+4 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — 3- 1+4⋊2C2
 Chief series C1 — C3 — C32 — He3 — C9○He3 — 3- 1+4 — 3- 1+4⋊2C2
 Lower central He3 — 3- 1+4⋊2C2
 Upper central C1 — C3 — 3- 1+2

Generators and relations for 3- 1+42C2
G = < a,b,c,d,e,f | a3=c3=d3=e3=f2=1, b3=a, cbc-1=ebe-1=ab=ba, dcd-1=ac=ca, ad=da, ae=ea, af=fa, bd=db, fbf=abce-1, ce=ec, fcf=ac-1e-1, de=ed, fdf=d-1e, ef=fe >

Subgroups: 504 in 184 conjugacy classes, 43 normal (10 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, S3×C9, He3⋊C2, C2×3- 1+2, S3×C32, C3×He3, C3×3- 1+2, C3×3- 1+2, C9○He3, C9○He3, S3×3- 1+2, C3×He3⋊C2, He3.4C6, 3- 1+4, 3- 1+42C2
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, S3×C32, C3×C3⋊S3, C32×C3⋊S3, 3- 1+42C2

Permutation representations of 3- 1+42C2
On 27 points - transitive group 27T167
Generators in S27
```(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)
(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)
(1 7 4)(2 5 8)(10 16 13)(11 14 17)(19 22 25)(21 27 24)
(1 18 19)(2 10 20)(3 11 21)(4 12 22)(5 13 23)(6 14 24)(7 15 25)(8 16 26)(9 17 27)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(20 26 23)(21 24 27)
(10 26)(11 24)(12 22)(13 20)(14 27)(15 25)(16 23)(17 21)(18 19)```

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

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

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

`G:=TransitiveGroup(27,167);`

58 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E ··· 3P 6A 6B 6C 6D 9A ··· 9F 9G ··· 9AD 18A ··· 18F order 1 2 3 3 3 3 3 ··· 3 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 1 3 3 6 ··· 6 9 9 27 27 3 ··· 3 6 ··· 6 27 ··· 27

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 9 type + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 3- 1+4⋊2C2 kernel 3- 1+4⋊2C2 3- 1+4 C3×He3⋊C2 He3.4C6 C3×He3 C9○He3 C3×3- 1+2 C3×C9 C33 C1 # reps 1 1 2 6 2 6 4 24 8 4

Matrix representation of 3- 1+42C2 in GL9(𝔽19)

 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7
,
 0 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 11 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0
,
 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 1
,
 0 0 1 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 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 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 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 11
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 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 0 0 1 0 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 0 0 1 0

`G:=sub<GL(9,GF(19))| [7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,0],[11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;`

3- 1+42C2 in GAP, Magma, Sage, TeX

`3_-^{1+4}\rtimes_2C_2`
`% in TeX`

`G:=Group("ES-(3,2):2C2");`
`// GroupNames label`

`G:=SmallGroup(486,239);`
`// by ID`

`G=gap.SmallGroup(486,239);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1520,500,867,3244,382]);`
`// Polycyclic`

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

׿
×
𝔽