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## G = C24⋊4S3order 96 = 25·3

### 1st semidirect product of C24 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24⋊4S3
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×C3⋊D4 — C24⋊4S3
 Lower central C3 — C2×C6 — C24⋊4S3
 Upper central C1 — C22 — C24

Generators and relations for C244S3
G = < a,b,c,d,e,f | a2=b2=c2=d2=e3=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 290 in 130 conjugacy classes, 41 normal (8 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×3], C22, C22 [×6], C22 [×17], S3, C6 [×3], C6 [×6], C2×C4 [×3], D4 [×6], C23 [×3], C23 [×7], Dic3 [×3], D6 [×3], C2×C6, C2×C6 [×6], C2×C6 [×14], C22⋊C4 [×3], C2×D4 [×3], C24, C2×Dic3 [×3], C3⋊D4 [×6], C22×S3, C22×C6 [×3], C22×C6 [×6], C22≀C2, C6.D4 [×3], C2×C3⋊D4 [×3], C23×C6, C244S3
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], C244S3

Character table of C244S3

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O size 1 1 1 1 2 2 2 2 2 2 12 2 12 12 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 -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 -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 1 1 linear of order 2 ρ5 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 1 1 linear of order 2 ρ6 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 -1 1 linear of order 2 ρ7 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 -1 1 linear of order 2 ρ8 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 1 1 linear of order 2 ρ9 2 2 -2 -2 0 0 -2 2 0 0 0 2 0 0 0 -2 -2 -2 2 2 0 0 0 0 0 0 2 0 0 -2 orthogonal lifted from D4 ρ10 2 -2 2 -2 -2 2 0 0 0 0 0 2 0 0 0 0 0 2 0 0 2 0 0 2 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ11 2 2 2 2 2 2 2 2 2 2 0 -1 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 -2 -2 2 2 -2 -2 0 -1 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ13 2 -2 -2 2 0 0 0 0 -2 2 0 2 0 0 0 0 0 -2 0 0 0 -2 -2 0 2 2 -2 0 0 2 orthogonal lifted from D4 ρ14 2 -2 -2 2 0 0 0 0 2 -2 0 2 0 0 0 0 0 -2 0 0 0 2 2 0 -2 -2 -2 0 0 2 orthogonal lifted from D4 ρ15 2 2 2 2 2 2 -2 -2 -2 -2 0 -1 0 0 0 1 1 -1 1 1 -1 1 1 -1 1 1 -1 -1 -1 -1 orthogonal lifted from D6 ρ16 2 2 2 2 -2 -2 -2 -2 2 2 0 -1 0 0 0 1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 orthogonal lifted from D6 ρ17 2 2 -2 -2 0 0 2 -2 0 0 0 2 0 0 0 2 2 -2 -2 -2 0 0 0 0 0 0 2 0 0 -2 orthogonal lifted from D4 ρ18 2 -2 2 -2 2 -2 0 0 0 0 0 2 0 0 0 0 0 2 0 0 -2 0 0 -2 0 0 -2 2 2 -2 orthogonal lifted from D4 ρ19 2 -2 -2 2 0 0 0 0 -2 2 0 -1 0 0 0 √-3 -√-3 1 -√-3 √-3 -√-3 1 1 √-3 -1 -1 1 √-3 -√-3 -1 complex lifted from C3⋊D4 ρ20 2 2 -2 -2 0 0 -2 2 0 0 0 -1 0 0 0 1 1 1 -1 -1 √-3 -√-3 √-3 -√-3 -√-3 √-3 -1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ21 2 -2 -2 2 0 0 0 0 2 -2 0 -1 0 0 0 √-3 -√-3 1 -√-3 √-3 √-3 -1 -1 -√-3 1 1 1 -√-3 √-3 -1 complex lifted from C3⋊D4 ρ22 2 -2 2 -2 -2 2 0 0 0 0 0 -1 0 0 0 √-3 -√-3 -1 √-3 -√-3 -1 √-3 -√-3 -1 -√-3 √-3 1 1 1 1 complex lifted from C3⋊D4 ρ23 2 2 -2 -2 0 0 2 -2 0 0 0 -1 0 0 0 -1 -1 1 1 1 √-3 √-3 -√-3 -√-3 √-3 -√-3 -1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ24 2 -2 2 -2 -2 2 0 0 0 0 0 -1 0 0 0 -√-3 √-3 -1 -√-3 √-3 -1 -√-3 √-3 -1 √-3 -√-3 1 1 1 1 complex lifted from C3⋊D4 ρ25 2 -2 2 -2 2 -2 0 0 0 0 0 -1 0 0 0 √-3 -√-3 -1 √-3 -√-3 1 -√-3 √-3 1 √-3 -√-3 1 -1 -1 1 complex lifted from C3⋊D4 ρ26 2 2 -2 -2 0 0 2 -2 0 0 0 -1 0 0 0 -1 -1 1 1 1 -√-3 -√-3 √-3 √-3 -√-3 √-3 -1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ27 2 -2 2 -2 2 -2 0 0 0 0 0 -1 0 0 0 -√-3 √-3 -1 -√-3 √-3 1 √-3 -√-3 1 -√-3 √-3 1 -1 -1 1 complex lifted from C3⋊D4 ρ28 2 -2 -2 2 0 0 0 0 2 -2 0 -1 0 0 0 -√-3 √-3 1 √-3 -√-3 -√-3 -1 -1 √-3 1 1 1 √-3 -√-3 -1 complex lifted from C3⋊D4 ρ29 2 2 -2 -2 0 0 -2 2 0 0 0 -1 0 0 0 1 1 1 -1 -1 -√-3 √-3 -√-3 √-3 √-3 -√-3 -1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ30 2 -2 -2 2 0 0 0 0 -2 2 0 -1 0 0 0 -√-3 √-3 1 √-3 -√-3 √-3 1 1 -√-3 -1 -1 1 -√-3 √-3 -1 complex lifted from C3⋊D4

Permutation representations of C244S3
On 24 points - transitive group 24T116
Generators in S24
```(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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 14)(2 13)(3 15)(4 17)(5 16)(6 18)(7 20)(8 19)(9 21)(10 23)(11 22)(12 24)```

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

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

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

`G:=TransitiveGroup(24,116);`

Matrix representation of C244S3 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 1
,
 1 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 9 0 0 0 0 3 0 0 0 0 3 0 0 0 0 9
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[9,0,0,0,0,3,0,0,0,0,3,0,0,0,0,9],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;`

C244S3 in GAP, Magma, Sage, TeX

`C_2^4\rtimes_4S_3`
`% in TeX`

`G:=Group("C2^4:4S3");`
`// GroupNames label`

`G:=SmallGroup(96,160);`
`// by ID`

`G=gap.SmallGroup(96,160);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,218,2309]);`
`// Polycyclic`

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

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