Copied to
clipboard

## G = C25.S3order 192 = 26·3

### 1st non-split extension by C25 of S3 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C25.S3
 Chief series C1 — C22 — A4 — C2×A4 — C22×A4 — C2×A4⋊C4 — C25.S3
 Lower central A4 — C2×A4 — C25.S3
 Upper central C1 — C22 — C23

Generators and relations for C25.S3
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f3=1, g2=b, ab=ba, gag-1=ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, fdf-1=gdg-1=e, geg-1=d, gfg-1=f-1 >

Subgroups: 714 in 173 conjugacy classes, 27 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C2×C4, C23, C23, C23, Dic3, A4, C2×C6, C22⋊C4, C22×C4, C24, C24, C24, C2×Dic3, C2×A4, C2×A4, C2×A4, C22×C6, C2×C22⋊C4, C25, C6.D4, A4⋊C4, C22×A4, C22×A4, C22×A4, C243C4, C2×A4⋊C4, C23×A4, C25.S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C2×Dic3, C3⋊D4, S4, C6.D4, A4⋊C4, C2×S4, C2×A4⋊C4, A4⋊D4, C25.S3

Character table of C25.S3

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G size 1 1 1 1 2 2 3 3 3 3 6 6 8 12 12 12 12 12 12 12 12 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ5 1 1 -1 -1 1 -1 -1 -1 1 1 -1 1 1 i i -i -i -i -i i i -1 -1 -1 1 1 1 -1 linear of order 4 ρ6 1 1 -1 -1 1 -1 -1 -1 1 1 -1 1 1 -i -i i i i i -i -i -1 -1 -1 1 1 1 -1 linear of order 4 ρ7 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 1 i -i i i -i -i i -i 1 1 -1 1 -1 -1 -1 linear of order 4 ρ8 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 1 -i i -i -i i i -i i 1 1 -1 1 -1 -1 -1 linear of order 4 ρ9 2 2 2 2 -2 -2 2 2 2 2 -2 -2 -1 0 0 0 0 0 0 0 0 1 1 -1 -1 1 1 -1 orthogonal lifted from D6 ρ10 2 -2 -2 2 0 0 -2 2 -2 2 0 0 2 0 0 0 0 0 0 0 0 0 0 2 -2 0 0 -2 orthogonal lifted from D4 ρ11 2 2 2 2 2 2 2 2 2 2 2 2 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 -2 2 -2 0 0 2 -2 -2 2 0 0 2 0 0 0 0 0 0 0 0 0 0 -2 -2 0 0 2 orthogonal lifted from D4 ρ13 2 2 -2 -2 -2 2 -2 -2 2 2 2 -2 -1 0 0 0 0 0 0 0 0 -1 -1 1 -1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 -2 -2 2 -2 -2 -2 2 2 -2 2 -1 0 0 0 0 0 0 0 0 1 1 1 -1 -1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 -2 2 0 0 -2 2 -2 2 0 0 -1 0 0 0 0 0 0 0 0 √-3 -√-3 -1 1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ16 2 -2 -2 2 0 0 -2 2 -2 2 0 0 -1 0 0 0 0 0 0 0 0 -√-3 √-3 -1 1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ17 2 -2 2 -2 0 0 2 -2 -2 2 0 0 -1 0 0 0 0 0 0 0 0 √-3 -√-3 1 1 √-3 -√-3 -1 complex lifted from C3⋊D4 ρ18 2 -2 2 -2 0 0 2 -2 -2 2 0 0 -1 0 0 0 0 0 0 0 0 -√-3 √-3 1 1 -√-3 √-3 -1 complex lifted from C3⋊D4 ρ19 3 3 3 3 3 3 -1 -1 -1 -1 -1 -1 0 -1 -1 1 -1 1 -1 1 1 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ20 3 3 3 3 -3 -3 -1 -1 -1 -1 1 1 0 -1 1 -1 1 1 -1 1 -1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ21 3 3 3 3 3 3 -1 -1 -1 -1 -1 -1 0 1 1 -1 1 -1 1 -1 -1 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ22 3 3 3 3 -3 -3 -1 -1 -1 -1 1 1 0 1 -1 1 -1 -1 1 -1 1 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ23 3 3 -3 -3 3 -3 1 1 -1 -1 1 -1 0 i i i -i i -i -i -i 0 0 0 0 0 0 0 complex lifted from A4⋊C4 ρ24 3 3 -3 -3 3 -3 1 1 -1 -1 1 -1 0 -i -i -i i -i i i i 0 0 0 0 0 0 0 complex lifted from A4⋊C4 ρ25 3 3 -3 -3 -3 3 1 1 -1 -1 -1 1 0 -i i i -i -i i i -i 0 0 0 0 0 0 0 complex lifted from A4⋊C4 ρ26 3 3 -3 -3 -3 3 1 1 -1 -1 -1 1 0 i -i -i i i -i -i i 0 0 0 0 0 0 0 complex lifted from A4⋊C4 ρ27 6 -6 -6 6 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4 ρ28 6 -6 6 -6 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4

Permutation representations of C25.S3
On 24 points - transitive group 24T402
Generators in S24
```(2 12)(4 10)(5 16)(7 14)(18 22)(20 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)
(1 11)(2 4)(3 9)(5 16)(6 15)(7 14)(8 13)(10 12)(17 19)(18 24)(20 22)(21 23)
(1 3)(2 12)(4 10)(5 14)(6 13)(7 16)(8 15)(9 11)(17 23)(18 20)(19 21)(22 24)
(1 19 6)(2 7 20)(3 17 8)(4 5 18)(9 21 15)(10 16 22)(11 23 13)(12 14 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)```

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

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

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

`G:=TransitiveGroup(24,402);`

On 24 points - transitive group 24T403
Generators in S24
```(1 3)(2 15)(4 13)(5 12)(6 8)(7 10)(9 11)(14 16)(17 24)(18 20)(19 22)(21 23)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 16)(2 13)(3 14)(4 15)(5 10)(6 11)(7 12)(8 9)(17 22)(18 23)(19 24)(20 21)
(2 13)(4 15)(5 10)(6 11)(7 12)(8 9)(18 23)(20 21)
(1 16)(3 14)(5 10)(6 11)(7 12)(8 9)(17 22)(19 24)
(1 9 23)(2 24 10)(3 11 21)(4 22 12)(5 13 19)(6 20 14)(7 15 17)(8 18 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)```

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

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

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

`G:=TransitiveGroup(24,403);`

On 24 points - transitive group 24T404
Generators in S24
```(1 23)(2 4)(3 21)(5 17)(6 8)(7 19)(9 16)(10 12)(11 14)(13 15)(18 20)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 21)(2 22)(3 23)(4 24)(5 19)(6 20)(7 17)(8 18)(9 14)(10 15)(11 16)(12 13)
(2 24)(4 22)(5 17)(7 19)(9 16)(10 13)(11 14)(12 15)
(1 23)(3 21)(6 18)(8 20)(9 16)(10 13)(11 14)(12 15)
(1 11 19)(2 20 12)(3 9 17)(4 18 10)(5 21 16)(6 13 22)(7 23 14)(8 15 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)```

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

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

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

`G:=TransitiveGroup(24,404);`

Matrix representation of C25.S3 in GL7(𝔽13)

 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 12 12 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0
,
 0 5 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0

`G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0],[0,5,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0] >;`

C25.S3 in GAP, Magma, Sage, TeX

`C_2^5.S_3`
`% in TeX`

`G:=Group("C2^5.S3");`
`// GroupNames label`

`G:=SmallGroup(192,991);`
`// by ID`

`G=gap.SmallGroup(192,991);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,28,141,1124,4037,285,2358,475]);`
`// Polycyclic`

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

Export

׿
×
𝔽