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G = 2+ 1+4.5S3order 192 = 26·3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — 2+ 1+4.5S3
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C23.23D6 — 2+ 1+4.5S3
 Lower central C3 — C6 — C22×C6 — 2+ 1+4.5S3
 Upper central C1 — C2 — C23 — 2+ 1+4

Generators and relations for 2+ 1+4.5S3
G = < a,b,c,d,e,f | a4=b2=d2=e3=1, c2=f2=a2, bab=a-1, ac=ca, ad=da, ae=ea, faf-1=a-1cd, fcf-1=bc=cb, fdf-1=bd=db, be=eb, bf=fb, dcd=a2c, ce=ec, de=ed, fef-1=e-1 >

Subgroups: 424 in 160 conjugacy classes, 43 normal (10 characteristic)
C1, C2, C2 [×6], C3, C4 [×7], C22 [×3], C22 [×8], C6, C6 [×6], C2×C4 [×3], C2×C4 [×9], D4 [×9], Q8, C23, C23 [×3], C23, Dic3 [×4], C12 [×3], C2×C6 [×3], C2×C6 [×8], C22⋊C4 [×6], C4⋊C4 [×3], C22×C4, C2×D4 [×3], C2×D4 [×3], C4○D4 [×3], C2×Dic3 [×6], C2×C12 [×3], C2×C12 [×3], C3×D4 [×9], C3×Q8, C22×C6, C22×C6 [×3], C22×C6, C23⋊C4 [×3], C22.D4 [×3], 2+ 1+4, Dic3⋊C4 [×3], C6.D4 [×3], C6.D4 [×3], C22×Dic3, C6×D4 [×3], C6×D4 [×3], C3×C4○D4 [×3], C23.7D4, C23.7D6 [×3], C23.23D6 [×3], C3×2+ 1+4, 2+ 1+4.5S3
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], C23.7D4, C244S3, 2+ 1+4.5S3

Smallest permutation representation of 2+ 1+4.5S3
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 3)(5 7)(10 12)(14 16)(18 20)(21 23)(25 27)(30 32)(33 35)(37 39)(42 44)(45 47)
(1 47 3 45)(2 48 4 46)(5 44 7 42)(6 41 8 43)(9 19 11 17)(10 20 12 18)(13 24 15 22)(14 21 16 23)(25 39 27 37)(26 40 28 38)(29 34 31 36)(30 35 32 33)
(1 42)(2 43)(3 44)(4 41)(5 45)(6 46)(7 47)(8 48)(9 29)(10 30)(11 31)(12 32)(13 28)(14 25)(15 26)(16 27)(17 34)(18 35)(19 36)(20 33)(21 37)(22 38)(23 39)(24 40)
(1 35 39)(2 36 40)(3 33 37)(4 34 38)(5 10 14)(6 11 15)(7 12 16)(8 9 13)(17 22 41)(18 23 42)(19 24 43)(20 21 44)(25 45 30)(26 46 31)(27 47 32)(28 48 29)
(1 42 3 44)(2 48 4 46)(5 47 7 45)(6 43 8 41)(9 22 11 24)(10 27 12 25)(13 17 15 19)(14 32 16 30)(18 37 20 39)(21 35 23 33)(26 36 28 34)(29 38 31 40)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B ··· 6J 12A ··· 12F order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 ··· 6 12 ··· 12 size 1 1 2 2 2 4 4 4 2 4 4 4 12 12 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 C23.7D4 2+ 1+4.5S3 kernel 2+ 1+4.5S3 C23.7D6 C23.23D6 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+4.5S3 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 12 11 0 0 0 0 1 0 0 0 0 12 0 0 0 0 1 1 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 12 12 0 1
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 5 8 8
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 8 0 0 0 0 0 8 8 5 10 0 0 0 5 8 8
,
 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 9 0 0 0 0 3 0 0 0 0 0 0 0 0 8 0 0 0 0 8 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,1,0,0,1,0,12,1,0,0,12,1,0,0,0,0,11,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,12,0,0,0,12,0,12,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,5,0,0,0,0,5,8,0,0,0,0,0,8],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,8,0,0,0,5,0,8,5,0,0,0,0,5,8,0,0,0,0,10,8],[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],[0,3,0,0,0,0,9,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;`

2+ 1+4.5S3 in GAP, Magma, Sage, TeX

`2_+^{1+4}._5S_3`
`% in TeX`

`G:=Group("ES+(2,2).5S3");`
`// GroupNames label`

`G:=SmallGroup(192,802);`
`// by ID`

`G=gap.SmallGroup(192,802);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,570,438,6278]);`
`// 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=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a^-1*c*d,f*c*f^-1=b*c=c*b,f*d*f^-1=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^-1=e^-1>;`
`// generators/relations`

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