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## G = 2+ 1+4⋊7S3order 192 = 26·3

### 2nd semidirect product of 2+ 1+4 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — 2+ 1+4⋊7S3
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C23⋊2D6 — 2+ 1+4⋊7S3
 Lower central C3 — C6 — C22×C6 — 2+ 1+4⋊7S3
 Upper central C1 — C2 — C23 — 2+ 1+4

Generators and relations for 2+ 1+47S3
G = < a,b,c,d,e,f | a4=b2=d2=e3=f2=1, c2=a2, bab=a-1, ac=ca, ad=da, ae=ea, faf=a-1cd, fcf=bc=cb, fdf=bd=db, be=eb, bf=fb, dcd=a2c, ce=ec, de=ed, fef=e-1 >

Subgroups: 680 in 198 conjugacy classes, 43 normal (10 characteristic)
C1, C2, C2 [×8], C3, C4 [×6], C22 [×3], C22 [×18], S3 [×2], C6, C6 [×6], C2×C4 [×3], C2×C4 [×6], D4 [×15], Q8, C23, C23 [×3], C23 [×6], Dic3 [×3], C12 [×3], D6 [×10], C2×C6 [×3], C2×C6 [×8], C22⋊C4 [×6], C2×D4 [×3], C2×D4 [×6], C4○D4 [×3], C24, C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C2×C12 [×3], C3×D4 [×9], C3×Q8, C22×S3 [×5], C22×C6, C22×C6 [×3], C22×C6, C23⋊C4 [×3], C22≀C2 [×3], 2+ 1+4, D6⋊C4 [×3], C6.D4 [×3], C2×C3⋊D4 [×3], C6×D4 [×3], C6×D4 [×3], C3×C4○D4 [×3], S3×C23, C2≀C22, C23.7D6 [×3], C232D6 [×3], C3×2+ 1+4, 2+ 1+47S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C2×C3⋊D4 [×3], C2≀C22, C244S3, 2+ 1+47S3

Permutation representations of 2+ 1+47S3
On 24 points - transitive group 24T334
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(5 7)(10 12)(13 15)(18 20)(21 23)
(1 22 3 24)(2 23 4 21)(5 20 7 18)(6 17 8 19)(9 14 11 16)(10 15 12 13)
(1 24)(2 21)(3 22)(4 23)(5 20)(6 17)(7 18)(8 19)(9 14)(10 15)(11 16)(12 13)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(5 12)(6 11)(7 10)(8 9)(13 18)(14 19)(15 20)(16 17)(21 23)```

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

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

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

`G:=TransitiveGroup(24,334);`

On 24 points - transitive group 24T349
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 8)(6 7)(9 10)(11 12)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)
(1 23 3 21)(2 24 4 22)(5 19 7 17)(6 20 8 18)(9 13 11 15)(10 14 12 16)
(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(1 3)(2 4)(5 11)(6 10)(7 9)(8 12)(13 20)(14 17)(15 18)(16 19)(21 22)(23 24)```

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

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

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

`G:=TransitiveGroup(24,349);`

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 6A 6B ··· 6J 12A ··· 12F order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 6 6 ··· 6 12 ··· 12 size 1 1 2 2 2 4 4 4 12 12 2 4 4 4 24 24 24 2 4 ··· 4 4 ··· 4

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 8 type + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D4 D6 C3⋊D4 C3⋊D4 C2≀C22 2+ 1+4⋊7S3 kernel 2+ 1+4⋊7S3 C23.7D6 C23⋊2D6 C3×2+ 1+4 2+ 1+4 C2×C12 C22×C6 C2×D4 C2×C4 C23 C3 C1 # reps 1 3 3 1 1 3 3 3 6 6 2 1

Matrix representation of 2+ 1+47S3 in GL6(𝔽13)

 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 1 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12 0 0 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
,
 3 0 0 0 0 0 0 9 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 1
,
 0 4 0 0 0 0 10 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 12 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1,0,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,10,0,0,0,0,4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;`

2+ 1+47S3 in GAP, Magma, Sage, TeX

`2_+^{1+4}\rtimes_7S_3`
`% in TeX`

`G:=Group("ES+(2,2):7S3");`
`// GroupNames label`

`G:=SmallGroup(192,803);`
`// by ID`

`G=gap.SmallGroup(192,803);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,570,438,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^4=b^2=d^2=e^3=f^2=1,c^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1*c*d,f*c*f=b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=a^2*c,c*e=e*c,d*e=e*d,f*e*f=e^-1>;`
`// generators/relations`

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