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## G = 3+ 1+4⋊2C2order 486 = 2·35

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

Aliases: 3+ 1+42C2, C336(C3×S3), (C3×He3)⋊12C6, (C3×He3)⋊18S3, He3.7(C3⋊S3), He3.14(C3×C6), He3⋊C24C32, C32.11(S3×C32), (C3×He3⋊C2)⋊5C3, C32.22(C3×C3⋊S3), C3.12(C32×C3⋊S3), SmallGroup(486,237)

Series: Derived Chief Lower central Upper central

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

Generators and relations for 3+ 1+42C2
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, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, gdg=d-1, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 1098 in 238 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C3, C3, S3, C6, C32, C32, C3×S3, C3×C6, He3, He3, C33, C33, He3⋊C2, C2×He3, S3×C32, C3×He3, C3×He3, S3×He3, C3×He3⋊C2, 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 27T169
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 8 26)(2 9 27)(3 7 25)(4 23 19)(5 24 20)(6 22 21)(10 15 18)(11 13 16)(12 14 17)
(2 9 27)(3 25 7)(4 23 19)(5 20 24)(10 15 18)(11 16 13)
(1 6 14)(2 4 15)(3 5 13)(7 24 16)(8 22 17)(9 23 18)(10 27 19)(11 25 20)(12 26 21)
(1 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)
(1 12 21)(2 10 19)(3 11 20)(4 9 15)(5 7 13)(6 8 14)(16 24 25)(17 22 26)(18 23 27)
(4 15)(5 13)(6 14)(10 19)(11 20)(12 21)(16 24)(17 22)(18 23)```

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

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

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

`G:=TransitiveGroup(27,169);`

58 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 3K ··· 3AT 6A 6B 6C ··· 6J order 1 2 3 3 3 ··· 3 3 ··· 3 6 6 6 ··· 6 size 1 9 1 1 3 ··· 3 6 ··· 6 9 9 27 ··· 27

58 irreducible representations

 dim 1 1 1 1 2 2 9 type + + + image C1 C2 C3 C6 S3 C3×S3 3+ 1+4⋊2C2 kernel 3+ 1+4⋊2C2 3+ 1+4 C3×He3⋊C2 C3×He3 C3×He3 C33 C1 # reps 1 1 8 8 4 32 4

Matrix representation of 3+ 1+42C2 in GL9(𝔽7)

 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 3 0 0 5 0 0 3 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 6 0 1 0 0 0 0 0 0 2 0 0 0 1 0 0 0 0 2 0 0 0 0 1 0 0 0 2 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
,
 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 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 1 0 0 6 0 0 2 0 0 1 0 0 6 0 0 0 2 0 1 0 0 6 0 0 0 0 2
,
 1 3 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 6 0 0 0 0 0 4 0 0 2 0 0 1 0 0 4 0 0 2 0 0 0 1 0 4 0 0 2 0 1 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 6 0 0 0 0 0 0 3 0 5 0 0 0 0 0 0 2 1 5 0 0 0 0 0 0 0 0 0 2 0 6 0 0 0 0 0 0 3 0 5 0 0 0 0 0 0 2 1 5 0 0 0 0 0 1 0 0 4 0 0 2 2 0 1 1 0 4 4 0 0 6 0 1 3 0 4 0 1 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 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(7))| [0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,6,6,2,2,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0],[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],[1,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,6,6,6,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,0,0,3,6,6,0,0,0,4,4,4,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,6,6,2,2,2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[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],[2,3,2,0,0,0,0,2,6,0,0,1,0,0,0,0,0,0,6,5,5,0,0,0,1,1,1,0,0,0,2,3,2,0,1,3,0,0,0,0,0,1,0,0,0,0,0,0,6,5,5,4,4,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;`

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(486,237);`
`// by ID`

`G=gap.SmallGroup(486,237);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1520,867,3244,382]);`
`// 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,a*g=g*a,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,c*g=g*c,d*e=e*d,g*d*g=d^-1,e*f=f*e,e*g=g*e,g*f*g=f^-1>;`
`// generators/relations`

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