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## G = C23.8S4order 192 = 26·3

### 2nd non-split extension by C23 of S4 acting via S4/C22=S3

Aliases: C23.8S4, C22.GL2(𝔽3), C2.C42⋊S3, C2.3(C42⋊S3), C23.3A44C2, SmallGroup(192,181)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2.C42 — C23.3A4 — C23.8S4
 Chief series C1 — C2 — C23 — C2.C42 — C23.3A4 — C23.8S4
 Lower central C23.3A4 — C23.8S4
 Upper central C1 — C2

Generators and relations for C23.8S4
G = < a,b,c,d,e,f,g | a2=b2=c2=f3=g2=1, d2=gag=fbf-1=abc, e2=faf-1=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, geg=be=eb, bg=gb, ede-1=cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=cde, gdg=de, fef-1=bcd, gfg=f-1 >

3C2
3C2
24C2
16C3
3C22
3C22
6C4
6C4
12C22
12C4
12C22
12C22
16S3
16C6
16S3
6C23
12D4
12D4
12C8
12C2×C4
12D4
4A4
16D6
4S4
4S4

Character table of C23.8S4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 6 8A 8B 8C 8D size 1 1 3 3 24 32 6 6 12 24 32 12 12 12 12 ρ1 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 linear of order 2 ρ3 2 2 2 2 0 -1 2 2 2 0 -1 0 0 0 0 orthogonal lifted from S3 ρ4 2 -2 2 -2 0 -1 0 0 0 0 1 -√-2 √-2 √-2 -√-2 complex lifted from GL2(𝔽3) ρ5 2 -2 2 -2 0 -1 0 0 0 0 1 √-2 -√-2 -√-2 √-2 complex lifted from GL2(𝔽3) ρ6 3 3 3 3 1 0 -1 -1 -1 1 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ7 3 3 3 3 -1 0 -1 -1 -1 -1 0 1 1 1 1 orthogonal lifted from S4 ρ8 3 3 -1 -1 1 0 -1+2i -1-2i 1 -1 0 -i -i i i complex lifted from C42⋊S3 ρ9 3 3 -1 -1 1 0 -1-2i -1+2i 1 -1 0 i i -i -i complex lifted from C42⋊S3 ρ10 3 3 -1 -1 -1 0 -1+2i -1-2i 1 1 0 i i -i -i complex lifted from C42⋊S3 ρ11 3 3 -1 -1 -1 0 -1-2i -1+2i 1 1 0 -i -i i i complex lifted from C42⋊S3 ρ12 4 -4 4 -4 0 1 0 0 0 0 -1 0 0 0 0 orthogonal lifted from GL2(𝔽3) ρ13 6 6 -2 -2 0 0 2 2 -2 0 0 0 0 0 0 orthogonal lifted from C42⋊S3 ρ14 6 -6 -2 2 0 0 0 0 0 0 0 √2 -√2 √2 -√2 orthogonal faithful ρ15 6 -6 -2 2 0 0 0 0 0 0 0 -√2 √2 -√2 √2 orthogonal faithful

Permutation representations of C23.8S4
On 24 points - transitive group 24T313
Generators in S24
```(9 15)(10 16)(11 13)(12 14)
(1 4)(2 3)(5 7)(6 8)
(1 4)(2 3)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)(17 19)(18 20)(21 23)(22 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 6 4 8)(2 7 3 5)(9 11)(10 14)(12 16)(13 15)(17 24)(18 23)(19 22)(20 21)
(1 12 17)(2 16 21)(3 10 23)(4 14 19)(5 11 22)(6 9 20)(7 13 24)(8 15 18)
(2 7)(3 5)(6 8)(9 18)(10 22)(11 23)(12 17)(13 21)(14 19)(15 20)(16 24)```

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

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

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

`G:=TransitiveGroup(24,313);`

On 24 points - transitive group 24T315
Generators in S24
```(3 4)(5 14)(6 13)(15 16)
(1 2)(7 8)(9 11)(10 12)
(1 2)(3 4)(5 14)(6 13)(7 8)(9 11)(10 12)(15 16)(17 19)(18 20)(21 23)(22 24)
(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 8 2 7)(3 16)(4 15)(6 13)(9 12 11 10)(17 18)(19 20)(22 24)
(1 14 20)(2 5 18)(3 24 10)(4 22 12)(6 19 8)(7 13 17)(9 16 23)(11 15 21)
(1 9)(2 11)(3 19)(4 17)(5 21)(6 24)(7 12)(8 10)(13 22)(14 23)(15 18)(16 20)```

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

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

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

`G:=TransitiveGroup(24,315);`

On 24 points - transitive group 24T426
Generators in S24
```(1 7)(2 8)(3 4)(5 6)(9 22)(10 23)(11 24)(12 21)
(1 2)(3 5)(4 6)(7 8)(13 19)(14 20)(15 17)(16 18)
(1 8)(2 7)(3 6)(4 5)(9 24)(10 21)(11 22)(12 23)(13 17)(14 18)(15 19)(16 20)
(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 4 2 6)(3 8 5 7)(9 24)(11 22)(13 18 19 16)(14 15 20 17)
(1 24 16)(2 11 14)(3 21 17)(4 23 15)(5 12 19)(6 10 13)(7 22 18)(8 9 20)
(1 16)(2 18)(3 17)(4 19)(5 15)(6 13)(7 14)(8 20)(11 22)(12 23)```

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

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

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

`G:=TransitiveGroup(24,426);`

On 24 points - transitive group 24T428
Generators in S24
```(1 3)(2 4)(5 7)(6 8)(13 21)(14 22)(15 23)(16 24)
(1 2)(3 4)(5 8)(6 7)(9 19)(10 20)(11 17)(12 18)
(1 4)(2 3)(5 6)(7 8)(9 17)(10 18)(11 19)(12 20)(13 23)(14 24)(15 21)(16 22)
(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 7 2 6)(3 5 4 8)(9 12 19 18)(10 17 20 11)(13 21)(14 16)(15 23)(22 24)
(1 22 11)(2 14 9)(3 24 17)(4 16 19)(5 15 20)(6 21 12)(7 13 10)(8 23 18)
(1 10)(2 20)(3 12)(4 18)(5 9)(6 17)(7 11)(8 19)(13 22)(14 15)(16 23)(21 24)```

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

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

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

`G:=TransitiveGroup(24,428);`

Matrix representation of C23.8S4 in GL5(𝔽73)

 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72
,
 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 1
,
 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 7 0 0 0 7 13 0 0 0 0 0 27 0 0 0 0 0 72 0 0 0 0 0 27
,
 7 13 0 0 0 13 66 0 0 0 0 0 27 0 0 0 0 0 46 0 0 0 0 0 1
,
 39 26 0 0 0 27 33 0 0 0 0 0 0 0 67 0 0 60 0 0 0 0 0 44 0
,
 68 36 0 0 0 48 5 0 0 0 0 0 0 45 0 0 0 13 0 0 0 0 0 0 72

`G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1],[72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,7,0,0,0,7,13,0,0,0,0,0,27,0,0,0,0,0,72,0,0,0,0,0,27],[7,13,0,0,0,13,66,0,0,0,0,0,27,0,0,0,0,0,46,0,0,0,0,0,1],[39,27,0,0,0,26,33,0,0,0,0,0,0,60,0,0,0,0,0,44,0,0,67,0,0],[68,48,0,0,0,36,5,0,0,0,0,0,0,13,0,0,0,45,0,0,0,0,0,0,72] >;`

C23.8S4 in GAP, Magma, Sage, TeX

`C_2^3._8S_4`
`% in TeX`

`G:=Group("C2^3.8S4");`
`// GroupNames label`

`G:=SmallGroup(192,181);`
`// by ID`

`G=gap.SmallGroup(192,181);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,57,254,1143,268,171,934,521,80,2524,2531,3540]);`
`// Polycyclic`

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

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