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## G = 2+ 1+4.4S3order 192 = 26·3

### 1st non-split extension by 2+ 1+4 of S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — 2+ 1+4.4S3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×Dic6 — Q8.14D6 — 2+ 1+4.4S3
 Lower central C3 — C6 — C2×C12 — 2+ 1+4.4S3
 Upper central C1 — C2 — C2×C4 — 2+ 1+4

Generators and relations for 2+ 1+4.4S3
G = < a,b,c,d,e,f | a4=b2=d2=e3=1, c2=f2=a2, bab=faf-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf-1=ab, dcd=fcf-1=a2c, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >

Subgroups: 392 in 152 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×4], C22, C22 [×8], C6, C6 [×5], C8 [×2], C2×C4, C2×C4 [×6], D4 [×2], D4 [×8], Q8 [×2], Q8 [×2], C23 [×2], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×8], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4, C2×D4 [×4], C2×Q8, C4○D4 [×2], C4○D4 [×2], C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×4], C3×D4 [×2], C3×D4 [×8], C3×Q8 [×2], C22×C6 [×2], C22×C6 [×2], C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ 1+4, C4.Dic3 [×2], C4×Dic3, D4.S3 [×2], C3⋊Q16 [×2], C6.D4 [×2], C2×Dic6, C6×D4, C6×D4 [×4], C3×C4○D4 [×2], C3×C4○D4 [×2], D4.9D4, C12.D4, Q83Dic3 [×2], C23.12D6, Q8.14D6 [×2], C3×2+ 1+4, 2+ 1+4.4S3
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], D4.9D4, C244S3, 2+ 1+4.4S3

Smallest permutation representation of 2+ 1+4.4S3
On 48 points
Generators in S48
```(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 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 45)(2 48)(3 47)(4 46)(5 19)(6 18)(7 17)(8 20)(9 15)(10 14)(11 13)(12 16)(21 34)(22 33)(23 36)(24 35)(25 43)(26 42)(27 41)(28 44)(29 39)(30 38)(31 37)(32 40)
(1 2 3 4)(5 8 7 6)(9 12 11 10)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 36 35 34)(37 40 39 38)(41 44 43 42)(45 48 47 46)
(1 35)(2 36)(3 33)(4 34)(5 28)(6 25)(7 26)(8 27)(9 32)(10 29)(11 30)(12 31)(13 38)(14 39)(15 40)(16 37)(17 42)(18 43)(19 44)(20 41)(21 46)(22 47)(23 48)(24 45)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 45)(6 11 46)(7 12 47)(8 9 48)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 42 37)(34 43 38)(35 44 39)(36 41 40)
(1 22 3 24)(2 21 4 23)(5 40 7 38)(6 39 8 37)(9 42 11 44)(10 41 12 43)(13 27 15 25)(14 26 16 28)(17 29 19 31)(18 32 20 30)(33 46 35 48)(34 45 36 47)```

`G:=sub<Sym(48)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,45)(2,48)(3,47)(4,46)(5,19)(6,18)(7,17)(8,20)(9,15)(10,14)(11,13)(12,16)(21,34)(22,33)(23,36)(24,35)(25,43)(26,42)(27,41)(28,44)(29,39)(30,38)(31,37)(32,40), (1,2,3,4)(5,8,7,6)(9,12,11,10)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,36,35,34)(37,40,39,38)(41,44,43,42)(45,48,47,46), (1,35)(2,36)(3,33)(4,34)(5,28)(6,25)(7,26)(8,27)(9,32)(10,29)(11,30)(12,31)(13,38)(14,39)(15,40)(16,37)(17,42)(18,43)(19,44)(20,41)(21,46)(22,47)(23,48)(24,45), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,45)(6,11,46)(7,12,47)(8,9,48)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,42,37)(34,43,38)(35,44,39)(36,41,40), (1,22,3,24)(2,21,4,23)(5,40,7,38)(6,39,8,37)(9,42,11,44)(10,41,12,43)(13,27,15,25)(14,26,16,28)(17,29,19,31)(18,32,20,30)(33,46,35,48)(34,45,36,47)>;`

`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)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,45)(2,48)(3,47)(4,46)(5,19)(6,18)(7,17)(8,20)(9,15)(10,14)(11,13)(12,16)(21,34)(22,33)(23,36)(24,35)(25,43)(26,42)(27,41)(28,44)(29,39)(30,38)(31,37)(32,40), (1,2,3,4)(5,8,7,6)(9,12,11,10)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,36,35,34)(37,40,39,38)(41,44,43,42)(45,48,47,46), (1,35)(2,36)(3,33)(4,34)(5,28)(6,25)(7,26)(8,27)(9,32)(10,29)(11,30)(12,31)(13,38)(14,39)(15,40)(16,37)(17,42)(18,43)(19,44)(20,41)(21,46)(22,47)(23,48)(24,45), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,45)(6,11,46)(7,12,47)(8,9,48)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,42,37)(34,43,38)(35,44,39)(36,41,40), (1,22,3,24)(2,21,4,23)(5,40,7,38)(6,39,8,37)(9,42,11,44)(10,41,12,43)(13,27,15,25)(14,26,16,28)(17,29,19,31)(18,32,20,30)(33,46,35,48)(34,45,36,47) );`

`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),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,45),(2,48),(3,47),(4,46),(5,19),(6,18),(7,17),(8,20),(9,15),(10,14),(11,13),(12,16),(21,34),(22,33),(23,36),(24,35),(25,43),(26,42),(27,41),(28,44),(29,39),(30,38),(31,37),(32,40)], [(1,2,3,4),(5,8,7,6),(9,12,11,10),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,36,35,34),(37,40,39,38),(41,44,43,42),(45,48,47,46)], [(1,35),(2,36),(3,33),(4,34),(5,28),(6,25),(7,26),(8,27),(9,32),(10,29),(11,30),(12,31),(13,38),(14,39),(15,40),(16,37),(17,42),(18,43),(19,44),(20,41),(21,46),(22,47),(23,48),(24,45)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,45),(6,11,46),(7,12,47),(8,9,48),(21,25,30),(22,26,31),(23,27,32),(24,28,29),(33,42,37),(34,43,38),(35,44,39),(36,41,40)], [(1,22,3,24),(2,21,4,23),(5,40,7,38),(6,39,8,37),(9,42,11,44),(10,41,12,43),(13,27,15,25),(14,26,16,28),(17,29,19,31),(18,32,20,30),(33,46,35,48),(34,45,36,47)])`

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 6A 6B ··· 6J 8A 8B 12A ··· 12F order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 6 6 ··· 6 8 8 12 ··· 12 size 1 1 2 4 4 4 4 2 2 2 4 4 12 12 24 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 D4.9D4 2+ 1+4.4S3 kernel 2+ 1+4.4S3 C12.D4 Q8⋊3Dic3 C23.12D6 Q8.14D6 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+4.4S3 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 0 0 0 0 0 0 27 0 0 0 0 0 0 46 0 0 0 0 0 0 27
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 27 0 0 0 0 46 0 0 0 0 27 0 0 0 0 46 0 0 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 46 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 46
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 46 0 0 46 0 0 0 0 0 0 27 0 0
,
 36 45 0 0 0 0 45 36 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
,
 36 45 0 0 0 0 28 37 0 0 0 0 0 0 0 72 0 0 0 0 1 0 0 0 0 0 0 0 0 46 0 0 0 0 46 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,27],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,46,0,0,0,0,27,0,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,27,0,0,0,0,0,0,46,0,0],[36,45,0,0,0,0,45,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,28,0,0,0,0,45,37,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,46,0,0,0,0,46,0] >;`

2+ 1+4.4S3 in GAP, Magma, Sage, TeX

`2_+^{1+4}._4S_3`
`% in TeX`

`G:=Group("ES+(2,2).4S3");`
`// GroupNames label`

`G:=SmallGroup(192,801);`
`// by ID`

`G=gap.SmallGroup(192,801);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,570,1684,851,438,102,6278]);`
`// Polycyclic`

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

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