Copied to
clipboard

## G = 3+ 1+4⋊3C2order 486 = 2·35

### 3rd semidirect product of 3+ 1+4 and C2 acting faithfully

Aliases: 3+ 1+43C2, He36(C3⋊S3), C334(C3⋊S3), (C3×He3)⋊19S3, C3.3(C34⋊C2), C32.8(C33⋊C2), SmallGroup(486,249)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — 3+ 1+4 — 3+ 1+4⋊3C2
 Chief series C1 — C3 — C32 — C33 — C3×He3 — 3+ 1+4 — 3+ 1+4⋊3C2
 Lower central 3+ 1+4 — 3+ 1+4⋊3C2
 Upper central C1 — C3

Generators and relations for 3+ 1+43C2
G = < a,b,c,d,e,f,g | a3=b3=c3=d3=f3=g2=1, e1=b-1, ab=ba, cac-1=ab-1, ad=da, ae=ea, af=fa, gag=a-1, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg=bc-1, de=ed, gdg=d-1, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 4666 in 586 conjugacy classes, 214 normal (4 characteristic)
C1, C2, C3, C3, S3, C6, C32, C32, C3×S3, C3⋊S3, He3, C33, He3⋊C2, C3×C3⋊S3, C3×He3, He35S3, 3+ 1+4, 3+ 1+43C2
Quotients: C1, C2, S3, C3⋊S3, C33⋊C2, C34⋊C2, 3+ 1+43C2

Permutation representations of 3+ 1+43C2
On 27 points - transitive group 27T174
Generators in S27
```(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 11 13)(2 12 14)(3 10 15)(4 8 27)(5 9 25)(6 7 26)(16 21 24)(17 19 22)(18 20 23)
(1 13 11)(3 10 15)(4 27 8)(6 7 26)(17 19 22)(18 23 20)
(1 20 4)(2 21 5)(3 19 6)(7 10 22)(8 11 23)(9 12 24)(13 18 27)(14 16 25)(15 17 26)
(1 13 11)(2 14 12)(3 15 10)(4 27 8)(5 25 9)(6 26 7)(16 24 21)(17 22 19)(18 23 20)
(1 27 18)(2 25 16)(3 26 17)(4 20 11)(5 21 12)(6 19 10)(7 22 15)(8 23 13)(9 24 14)
(2 3)(4 20)(5 19)(6 21)(7 24)(8 23)(9 22)(10 12)(14 15)(16 26)(17 25)(18 27)```

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

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

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

`G:=TransitiveGroup(27,174);`

46 conjugacy classes

 class 1 2 3A 3B 3C ··· 3AP 6A 6B order 1 2 3 3 3 ··· 3 6 6 size 1 81 1 1 6 ··· 6 81 81

46 irreducible representations

 dim 1 1 2 9 type + + + image C1 C2 S3 3+ 1+4⋊3C2 kernel 3+ 1+4⋊3C2 3+ 1+4 C3×He3 C1 # reps 1 1 40 4

Matrix representation of 3+ 1+43C2 in GL9(𝔽7)

 0 0 4 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0
,
 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4
,
 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4
,
 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 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0
,
 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2
,
 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 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 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2 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 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 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0

`G:=sub<GL(9,GF(7))| [0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4],[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],[2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2],[0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0] >;`

3+ 1+43C2 in GAP, Magma, Sage, TeX

`3_+^{1+4}\rtimes_3C_2`
`% in TeX`

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

`G:=SmallGroup(486,249);`
`// by ID`

`G=gap.SmallGroup(486,249);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,218,867,735,3244,3250]);`
`// Polycyclic`

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

׿
×
𝔽