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## G = C23.7S4order 192 = 26·3

### 1st non-split extension by C23 of S4 acting via S4/C22=S3

Aliases: C23.7S4, C22.CSU2(𝔽3), C2.2(C42⋊S3), C2.C42.S3, C23.3A4.2C2, SmallGroup(192,180)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2.C42 — C23.3A4 — C23.7S4
 Chief series C1 — C2 — C23 — C2.C42 — C23.3A4 — C23.7S4
 Lower central C23.3A4 — C23.7S4
 Upper central C1 — C2

Generators and relations for C23.7S4
G = < a,b,c,d,e,f,g | a2=b2=c2=f3=1, d2=gag-1=fbf-1=abc, e2=faf-1=b, g2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, geg-1=be=eb, bg=gb, ede-1=cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=cde, gdg-1=de, fef-1=bcd, gfg-1=f-1 >

3C2
3C2
16C3
3C22
3C22
6C4
6C4
12C4
12C4
16C6
12C2×C4
12Q8
12C8
4A4
16Dic3

Character table of C23.7S4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6 8A 8B 8C 8D size 1 1 3 3 32 6 6 12 24 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 -1 2 2 2 0 0 -1 0 0 0 0 orthogonal lifted from S3 ρ4 2 -2 -2 2 -1 0 0 0 0 0 1 -√2 √2 √2 -√2 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ5 2 -2 -2 2 -1 0 0 0 0 0 1 √2 -√2 -√2 √2 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ6 3 3 3 3 0 -1 -1 -1 -1 -1 0 1 1 1 1 orthogonal lifted from S4 ρ7 3 3 3 3 0 -1 -1 -1 1 1 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ8 3 3 -1 -1 0 -1+2i -1-2i 1 1 -1 0 i i -i -i complex lifted from C42⋊S3 ρ9 3 3 -1 -1 0 -1-2i -1+2i 1 -1 1 0 i i -i -i complex lifted from C42⋊S3 ρ10 3 3 -1 -1 0 -1+2i -1-2i 1 -1 1 0 -i -i i i complex lifted from C42⋊S3 ρ11 3 3 -1 -1 0 -1-2i -1+2i 1 1 -1 0 -i -i i i complex lifted from C42⋊S3 ρ12 4 -4 -4 4 1 0 0 0 0 0 -1 0 0 0 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ13 6 6 -2 -2 0 2 2 -2 0 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 complex faithful ρ15 6 -6 2 -2 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 complex faithful

Permutation representations of C23.7S4
On 24 points - transitive group 24T316
Generators in S24
```(3 4)(9 13)(10 14)(15 16)
(1 2)(5 6)(7 11)(8 12)
(1 2)(3 4)(5 6)(7 11)(8 12)(9 13)(10 14)(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 5 2 6)(3 15)(4 16)(7 8 11 12)(10 14)(17 18)(19 20)(21 23)
(1 9 18)(2 13 20)(3 23 7)(4 21 11)(5 14 17)(6 10 19)(8 15 22)(12 16 24)
(1 8 2 12)(3 19 4 17)(5 7 6 11)(9 22 13 24)(10 21 14 23)(15 20 16 18)```

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

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

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

`G:=TransitiveGroup(24,316);`

On 24 points - transitive group 24T427
Generators in S24
```(1 2)(3 5)(4 6)(7 8)(17 21)(18 22)(19 23)(20 24)
(1 7)(2 8)(3 4)(5 6)(9 16)(10 13)(11 14)(12 15)
(1 8)(2 7)(3 6)(4 5)(9 14)(10 15)(11 16)(12 13)(17 23)(18 24)(19 21)(20 22)
(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 3 7 4)(2 5 8 6)(9 10 16 13)(11 12 14 15)(18 24)(20 22)
(1 19 10)(2 17 13)(3 24 9)(4 20 11)(5 22 16)(6 18 14)(7 23 12)(8 21 15)
(1 13 8 12)(2 15 7 10)(3 16 6 11)(4 9 5 14)(17 21 23 19)(18 20 24 22)```

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

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

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

`G:=TransitiveGroup(24,427);`

On 24 points - transitive group 24T429
Generators in S24
```(1 4)(2 3)(5 8)(6 7)(13 20)(14 17)(15 18)(16 19)
(1 3)(2 4)(5 7)(6 8)(9 22)(10 23)(11 24)(12 21)
(1 2)(3 4)(5 6)(7 8)(9 24)(10 21)(11 22)(12 23)(13 18)(14 19)(15 20)(16 17)
(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 7 3 5)(2 8 4 6)(9 23 22 10)(11 21 24 12)(13 20)(14 16)(15 18)(17 19)
(1 14 10)(2 19 21)(3 17 12)(4 16 23)(5 13 11)(6 18 22)(7 20 9)(8 15 24)
(1 22 2 11)(3 9 4 24)(5 10 6 21)(7 23 8 12)(13 14 18 19)(15 17 20 16)```

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

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

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

`G:=TransitiveGroup(24,429);`

Matrix representation of C23.7S4 in GL5(𝔽73)

 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 72 0 0 0 0 0 72
,
 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
,
 66 17 0 0 0 40 7 0 0 0 0 0 46 0 0 0 0 0 72 0 0 0 0 0 46
,
 27 35 0 0 0 0 46 0 0 0 0 0 1 0 0 0 0 0 27 0 0 0 0 0 46
,
 11 34 0 0 0 24 61 0 0 0 0 0 0 46 0 0 0 0 0 1 0 0 27 0 0
,
 65 2 0 0 0 4 8 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

`G:=sub<GL(5,GF(73))| [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],[72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[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],[66,40,0,0,0,17,7,0,0,0,0,0,46,0,0,0,0,0,72,0,0,0,0,0,46],[27,0,0,0,0,35,46,0,0,0,0,0,1,0,0,0,0,0,27,0,0,0,0,0,46],[11,24,0,0,0,34,61,0,0,0,0,0,0,0,27,0,0,46,0,0,0,0,0,1,0],[65,4,0,0,0,2,8,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;`

C23.7S4 in GAP, Magma, Sage, TeX

`C_2^3._7S_4`
`% in TeX`

`G:=Group("C2^3.7S4");`
`// GroupNames label`

`G:=SmallGroup(192,180);`
`// by ID`

`G=gap.SmallGroup(192,180);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,57,254,1143,268,171,934,521,80,2524,2531,3540]);`
`// Polycyclic`

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

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