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## G = 2+ 1+4⋊6S3order 192 = 26·3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — 2+ 1+4⋊6S3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — D4⋊D6 — 2+ 1+4⋊6S3
 Lower central C3 — C6 — C2×C12 — 2+ 1+4⋊6S3
 Upper central C1 — C2 — C2×C4 — 2+ 1+4

Generators and relations for 2+ 1+46S3
G = < a,b,c,d,e,f | a4=b2=d2=e3=f2=1, c2=a2, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, dcd=fcf=a2c, ce=ec, de=ed, fdf=cd, fef=e-1 >

Subgroups: 520 in 168 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C4○D4, C3⋊C8, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C22×C6, C4.D4, C4≀C2, C41D4, C8⋊C22, 2+ 1+4, C4.Dic3, C4×Dic3, D4⋊S3, Q82S3, C2×D12, C2×C3⋊D4, C6×D4, C6×D4, C3×C4○D4, C3×C4○D4, D44D4, C12.D4, Q83Dic3, C123D4, D4⋊D6, C3×2+ 1+4, 2+ 1+46S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×C3⋊D4, D44D4, C244S3, 2+ 1+46S3

Permutation representations of 2+ 1+46S3
On 24 points - transitive group 24T347
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 24)(2 23)(3 22)(4 21)(5 13)(6 16)(7 15)(8 14)(9 19)(10 18)(11 17)(12 20)
(1 2 3 4)(5 8 7 6)(9 12 11 10)(13 14 15 16)(17 18 19 20)(21 24 23 22)
(1 23)(2 24)(3 21)(4 22)(5 16)(6 13)(7 14)(8 15)(9 20)(10 17)(11 18)(12 19)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(2 4)(5 11)(6 10)(7 9)(8 12)(13 20)(14 19)(15 18)(16 17)(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), (1,24)(2,23)(3,22)(4,21)(5,13)(6,16)(7,15)(8,14)(9,19)(10,18)(11,17)(12,20), (1,2,3,4)(5,8,7,6)(9,12,11,10)(13,14,15,16)(17,18,19,20)(21,24,23,22), (1,23)(2,24)(3,21)(4,22)(5,16)(6,13)(7,14)(8,15)(9,20)(10,17)(11,18)(12,19), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (2,4)(5,11)(6,10)(7,9)(8,12)(13,20)(14,19)(15,18)(16,17)(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), (1,24)(2,23)(3,22)(4,21)(5,13)(6,16)(7,15)(8,14)(9,19)(10,18)(11,17)(12,20), (1,2,3,4)(5,8,7,6)(9,12,11,10)(13,14,15,16)(17,18,19,20)(21,24,23,22), (1,23)(2,24)(3,21)(4,22)(5,16)(6,13)(7,14)(8,15)(9,20)(10,17)(11,18)(12,19), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (2,4)(5,11)(6,10)(7,9)(8,12)(13,20)(14,19)(15,18)(16,17)(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)], [(1,24),(2,23),(3,22),(4,21),(5,13),(6,16),(7,15),(8,14),(9,19),(10,18),(11,17),(12,20)], [(1,2,3,4),(5,8,7,6),(9,12,11,10),(13,14,15,16),(17,18,19,20),(21,24,23,22)], [(1,23),(2,24),(3,21),(4,22),(5,16),(6,13),(7,14),(8,15),(9,20),(10,17),(11,18),(12,19)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(2,4),(5,11),(6,10),(7,9),(8,12),(13,20),(14,19),(15,18),(16,17),(21,22),(23,24)]])`

`G:=TransitiveGroup(24,347);`

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 6A 6B ··· 6J 8A 8B 12A ··· 12F order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 6 6 ··· 6 8 8 12 ··· 12 size 1 1 2 4 4 4 4 24 2 2 2 4 4 12 12 2 4 ··· 4 24 24 4 ··· 4

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 8 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 C3⋊D4 C3⋊D4 C3⋊D4 D4⋊4D4 2+ 1+4⋊6S3 kernel 2+ 1+4⋊6S3 C12.D4 Q8⋊3Dic3 C12⋊3D4 D4⋊D6 C3×2+ 1+4 2+ 1+4 C3×D4 C3×Q8 C22×C6 C2×D4 C4○D4 D4 Q8 C23 C3 C1 # reps 1 1 2 1 2 1 1 2 2 2 1 2 4 4 4 2 1

Matrix representation of 2+ 1+46S3 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 2 0 0 0 0 72 1 0 0 0 0 0 0 72 2 0 0 0 0 72 1
,
 1 72 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 2 0 0 0 0 0 1 0 0 72 2 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 71 0 0 0 0 1 72 0 0 0 0 0 0 72 2 0 0 0 0 72 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 2 0 0 0 0 72 1 0 0 1 71 0 0 0 0 1 72 0 0
,
 64 45 0 0 0 0 0 8 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
,
 1 0 0 0 0 0 2 72 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 1 71 0 0 0 0 0 72

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,2,1,0,0,72,0,0,0,0,0,2,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,72,72,0,0,0,0,2,1,0,0],[64,0,0,0,0,0,45,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,71,72] >;`

2+ 1+46S3 in GAP, Magma, Sage, TeX

`2_+^{1+4}\rtimes_6S_3`
`% in TeX`

`G:=Group("ES+(2,2):6S3");`
`// GroupNames label`

`G:=SmallGroup(192,800);`
`// by ID`

`G=gap.SmallGroup(192,800);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,570,1684,851,438,102,6278]);`
`// 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=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a*b,d*c*d=f*c*f=a^2*c,c*e=e*c,d*e=e*d,f*d*f=c*d,f*e*f=e^-1>;`
`// generators/relations`

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