Copied to
clipboard

## G = 2- 1+4.2S3order 192 = 26·3

### The 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+4.2S3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×Dic6 — Q8.14D6 — 2- 1+4.2S3
 Lower central C3 — C6 — C2×C12 — 2- 1+4.2S3
 Upper central C1 — C2 — C2×C4 — 2- 1+4

Generators and relations for 2- 1+4.2S3
G = < a,b,c,d,e,f | a4=b2=e3=1, c2=d2=f2=a2, bab=faf-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf-1=ab, dcd-1=fcf-1=a2c, ce=ec, de=ed, fdf-1=a2cd, fef-1=e-1 >

Subgroups: 328 in 142 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C4.Dic3, C4×Dic3, Dic3⋊C4, D4.S3, C3⋊Q16, C2×Dic6, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, D4.10D4, C12.10D4, Q83Dic3, Dic3⋊Q8, Q8.14D6, C3×2- 1+4, 2- 1+4.2S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×C3⋊D4, D4.10D4, C244S3, 2- 1+4.2S3

Smallest permutation representation of 2- 1+4.2S3
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 33)(2 36)(3 35)(4 34)(5 31)(6 30)(7 29)(8 32)(9 27)(10 26)(11 25)(12 28)(13 43)(14 42)(15 41)(16 44)(17 39)(18 38)(19 37)(20 40)(21 46)(22 45)(23 48)(24 47)
(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 47 3 45)(2 48 4 46)(5 17 7 19)(6 18 8 20)(9 15 11 13)(10 16 12 14)(21 36 23 34)(22 33 24 35)(25 43 27 41)(26 44 28 42)(29 37 31 39)(30 38 32 40)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 22 3 24)(2 21 4 23)(5 41 7 43)(6 44 8 42)(9 37 11 39)(10 40 12 38)(13 30 15 32)(14 29 16 31)(17 26 19 28)(18 25 20 27)(33 48 35 46)(34 47 36 45)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B ··· 6F 8A 8B 12A ··· 12J order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 ··· 6 8 8 12 ··· 12 size 1 1 2 4 4 2 2 2 4 4 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 D4.10D4 2- 1+4.2S3 kernel 2- 1+4.2S3 C12.10D4 Q8⋊3Dic3 Dic3⋊Q8 Q8.14D6 C3×2- 1+4 2- 1+4 C2×C12 C3×D4 C3×Q8 C2×Q8 C4○D4 C2×C4 D4 Q8 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+4.2S3 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 72 72 1 2 0 0 0 1 72 72
,
 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 1 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 72 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 1 1 72 71 0 0 72 0 1 1
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 41 41 15 64 0 0 17 17 24 39 0 0 56 41 0 0 0 0 17 0 56 15
,
 64 0 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
,
 0 8 0 0 0 0 64 0 0 0 0 0 0 0 41 17 0 0 0 0 17 32 0 0 0 0 41 41 15 64 0 0 0 32 17 58

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,1,0,72,1,0,0,0,0,1,72,0,0,0,0,2,72],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,72,0,1,72,0,0,1,1,0,1,0,0,2,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,1,72,0,0,1,0,1,0,0,0,0,0,72,1,0,0,0,0,71,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,41,17,56,17,0,0,41,17,41,0,0,0,15,24,0,56,0,0,64,39,0,15],[64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,64,0,0,0,0,8,0,0,0,0,0,0,0,41,17,41,0,0,0,17,32,41,32,0,0,0,0,15,17,0,0,0,0,64,58] >;`

2- 1+4.2S3 in GAP, Magma, Sage, TeX

`2_-^{1+4}._2S_3`
`% in TeX`

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

`G:=SmallGroup(192,805);`
`// by ID`

`G=gap.SmallGroup(192,805);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,184,570,1684,851,438,102,6278]);`
`// Polycyclic`

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

׿
×
𝔽