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## G = C4.19S3≀C2order 288 = 25·32

### 4th central extension by C4 of S3≀C2

Aliases: C4.19S3≀C2, (C3×C12).19D4, C322C8.4C4, C321(C8.C4), C12.31D6.2C2, (C2×C3⋊S3).1Q8, (C3×C6).2(C4⋊C4), C3⋊S33C8.3C2, C3⋊Dic3.8(C2×C4), C2.3(C3⋊S3.Q8), (C4×C3⋊S3).53C22, SmallGroup(288,381)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C4.19S3≀C2
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C4×C3⋊S3 — C12.31D6 — C4.19S3≀C2
 Lower central C32 — C3×C6 — C3⋊Dic3 — C4.19S3≀C2
 Upper central C1 — C4

Generators and relations for C4.19S3≀C2
G = < a,b,c,d,e | a4=b3=c3=1, d4=a2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe-1=dcd-1=b-1, ce=ec, ede-1=d3 >

Character table of C4.19S3≀C2

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 18 4 4 1 1 18 4 4 12 12 12 12 18 18 18 18 4 4 4 4 12 12 12 12 12 12 12 12 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 i i -i -i 1 -1 1 -1 -1 -1 -1 -1 i i i i -i -i -i -i linear of order 4 ρ6 1 1 -1 1 1 -1 -1 1 1 1 i -i -i i -1 1 -1 1 -1 -1 -1 -1 i -i -i i -i i i -i linear of order 4 ρ7 1 1 -1 1 1 -1 -1 1 1 1 -i -i i i 1 -1 1 -1 -1 -1 -1 -1 -i -i -i -i i i i i linear of order 4 ρ8 1 1 -1 1 1 -1 -1 1 1 1 -i i i -i -1 1 -1 1 -1 -1 -1 -1 -i i i -i i -i -i i linear of order 4 ρ9 2 2 -2 2 2 2 2 -2 2 2 0 0 0 0 0 0 0 0 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 2 -2 0 2 2 2i -2i 0 -2 -2 0 0 0 0 √2 √-2 -√2 -√-2 2i -2i -2i 2i 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ12 2 -2 0 2 2 -2i 2i 0 -2 -2 0 0 0 0 √2 -√-2 -√2 √-2 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ13 2 -2 0 2 2 -2i 2i 0 -2 -2 0 0 0 0 -√2 √-2 √2 -√-2 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ14 2 -2 0 2 2 2i -2i 0 -2 -2 0 0 0 0 -√2 -√-2 √2 √-2 2i -2i -2i 2i 0 0 0 0 0 0 0 0 complex lifted from C8.C4 ρ15 4 4 0 -2 1 4 4 0 1 -2 0 -2 0 -2 0 0 0 0 1 1 -2 -2 0 1 1 0 0 1 1 0 orthogonal lifted from S3≀C2 ρ16 4 4 0 1 -2 4 4 0 -2 1 -2 0 -2 0 0 0 0 0 -2 -2 1 1 1 0 0 1 1 0 0 1 orthogonal lifted from S3≀C2 ρ17 4 4 0 -2 1 4 4 0 1 -2 0 2 0 2 0 0 0 0 1 1 -2 -2 0 -1 -1 0 0 -1 -1 0 orthogonal lifted from S3≀C2 ρ18 4 4 0 1 -2 4 4 0 -2 1 2 0 2 0 0 0 0 0 -2 -2 1 1 -1 0 0 -1 -1 0 0 -1 orthogonal lifted from S3≀C2 ρ19 4 4 0 -2 1 -4 -4 0 1 -2 0 -2i 0 2i 0 0 0 0 -1 -1 2 2 0 i i 0 0 -i -i 0 complex lifted from C3⋊S3.Q8 ρ20 4 4 0 1 -2 -4 -4 0 -2 1 -2i 0 2i 0 0 0 0 0 2 2 -1 -1 i 0 0 i -i 0 0 -i complex lifted from C3⋊S3.Q8 ρ21 4 4 0 1 -2 -4 -4 0 -2 1 2i 0 -2i 0 0 0 0 0 2 2 -1 -1 -i 0 0 -i i 0 0 i complex lifted from C3⋊S3.Q8 ρ22 4 4 0 -2 1 -4 -4 0 1 -2 0 2i 0 -2i 0 0 0 0 -1 -1 2 2 0 -i -i 0 0 i i 0 complex lifted from C3⋊S3.Q8 ρ23 4 -4 0 -2 1 4i -4i 0 -1 2 0 0 0 0 0 0 0 0 i -i 2i -2i 0 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 0 0 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 0 complex faithful ρ24 4 -4 0 -2 1 -4i 4i 0 -1 2 0 0 0 0 0 0 0 0 -i i -2i 2i 0 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 0 0 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 0 complex faithful ρ25 4 -4 0 -2 1 -4i 4i 0 -1 2 0 0 0 0 0 0 0 0 -i i -2i 2i 0 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 0 0 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 0 complex faithful ρ26 4 -4 0 1 -2 4i -4i 0 2 -1 0 0 0 0 0 0 0 0 -2i 2i -i i 2ζ85ζ3+ζ85 0 0 2ζ8ζ3+ζ8 2ζ87ζ3+ζ87 0 0 2ζ83ζ3+ζ83 complex faithful ρ27 4 -4 0 1 -2 -4i 4i 0 2 -1 0 0 0 0 0 0 0 0 2i -2i i -i 2ζ87ζ3+ζ87 0 0 2ζ83ζ3+ζ83 2ζ85ζ3+ζ85 0 0 2ζ8ζ3+ζ8 complex faithful ρ28 4 -4 0 1 -2 -4i 4i 0 2 -1 0 0 0 0 0 0 0 0 2i -2i i -i 2ζ83ζ3+ζ83 0 0 2ζ87ζ3+ζ87 2ζ8ζ3+ζ8 0 0 2ζ85ζ3+ζ85 complex faithful ρ29 4 -4 0 1 -2 4i -4i 0 2 -1 0 0 0 0 0 0 0 0 -2i 2i -i i 2ζ8ζ3+ζ8 0 0 2ζ85ζ3+ζ85 2ζ83ζ3+ζ83 0 0 2ζ87ζ3+ζ87 complex faithful ρ30 4 -4 0 -2 1 4i -4i 0 -1 2 0 0 0 0 0 0 0 0 i -i 2i -2i 0 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 0 0 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 0 complex faithful

Smallest permutation representation of C4.19S3≀C2
On 48 points
Generators in S48
```(1 7 5 3)(2 8 6 4)(9 46 13 42)(10 47 14 43)(11 48 15 44)(12 41 16 45)(17 19 21 23)(18 20 22 24)(25 37 29 33)(26 38 30 34)(27 39 31 35)(28 40 32 36)
(2 47 16)(4 10 41)(6 43 12)(8 14 45)(17 38 28)(19 30 40)(21 34 32)(23 26 36)
(1 46 15)(3 9 48)(5 42 11)(7 13 44)(18 29 39)(20 33 31)(22 25 35)(24 37 27)
(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 20 7 22 5 24 3 18)(2 23 8 17 6 19 4 21)(9 29 46 33 13 25 42 37)(10 32 47 36 14 28 43 40)(11 27 48 39 15 31 44 35)(12 30 41 34 16 26 45 38)```

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

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

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

Matrix representation of C4.19S3≀C2 in GL4(𝔽5) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 2 0 2 0 0 2 0 1 4 0 2 0 0 3 0 2
,
 2 0 2 0 0 2 0 4 4 0 2 0 0 2 0 2
,
 0 0 0 3 0 0 4 0 0 3 0 0 1 0 0 0
,
 0 3 0 0 1 0 0 0 0 0 0 1 0 0 3 0
`G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[2,0,4,0,0,2,0,3,2,0,2,0,0,1,0,2],[2,0,4,0,0,2,0,2,2,0,2,0,0,4,0,2],[0,0,0,1,0,0,3,0,0,4,0,0,3,0,0,0],[0,1,0,0,3,0,0,0,0,0,0,3,0,0,1,0] >;`

C4.19S3≀C2 in GAP, Magma, Sage, TeX

`C_4._{19}S_3\wr C_2`
`% in TeX`

`G:=Group("C4.19S3wrC2");`
`// GroupNames label`

`G:=SmallGroup(288,381);`
`// by ID`

`G=gap.SmallGroup(288,381);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,85,64,422,219,100,80,2693,2028,691,797,2372]);`
`// Polycyclic`

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

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