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## G = C4⋊C4⋊S3order 96 = 25·3

### 6th semidirect product of C4⋊C4 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C4⋊C4⋊S3
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — D6⋊C4 — C4⋊C4⋊S3
 Lower central C3 — C2×C6 — C4⋊C4⋊S3
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C4⋊C4⋊S3
G = < a,b,c,d | a4=b4=c3=d2=1, bab-1=a-1, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 138 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×Dic3, C2×C12, C22×S3, C422C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C4⋊C4⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, C422C2, C4○D12, D42S3, Q83S3, C4⋊C4⋊S3

Character table of C4⋊C4⋊S3

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 1 1 12 2 2 2 4 4 6 6 6 6 12 2 2 2 4 4 4 4 4 4 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 0 -1 -2 -2 2 -2 0 0 0 0 0 -1 -1 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 0 -1 -2 -2 -2 2 0 0 0 0 0 -1 -1 -1 -1 1 1 1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 0 -1 2 2 -2 -2 0 0 0 0 0 -1 -1 -1 1 -1 1 1 -1 1 orthogonal lifted from D6 ρ12 2 2 2 2 0 -1 2 2 2 2 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 -2 -2 2 0 2 0 0 0 0 -2i 0 2i 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 -2 2 -2 0 2 0 0 0 0 0 2i 0 -2i 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 2 2 -2 -2 0 2 2i -2i 0 0 0 0 0 0 0 -2 2 -2 0 2i 0 0 -2i 0 complex lifted from C4○D4 ρ16 2 -2 2 -2 0 2 0 0 0 0 0 -2i 0 2i 0 2 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ17 2 2 -2 -2 0 2 -2i 2i 0 0 0 0 0 0 0 -2 2 -2 0 -2i 0 0 2i 0 complex lifted from C4○D4 ρ18 2 -2 -2 2 0 2 0 0 0 0 2i 0 -2i 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 2 -2 -2 0 -1 2i -2i 0 0 0 0 0 0 0 1 -1 1 -√-3 -i √3 -√3 i √-3 complex lifted from C4○D12 ρ20 2 2 -2 -2 0 -1 -2i 2i 0 0 0 0 0 0 0 1 -1 1 -√-3 i -√3 √3 -i √-3 complex lifted from C4○D12 ρ21 2 2 -2 -2 0 -1 2i -2i 0 0 0 0 0 0 0 1 -1 1 √-3 -i -√3 √3 i -√-3 complex lifted from C4○D12 ρ22 2 2 -2 -2 0 -1 -2i 2i 0 0 0 0 0 0 0 1 -1 1 √-3 i √3 -√3 -i -√-3 complex lifted from C4○D12 ρ23 4 -4 -4 4 0 -2 0 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ24 4 -4 4 -4 0 -2 0 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2

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

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

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

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

Matrix representation of C4⋊C4⋊S3 in GL4(𝔽13) generated by

 11 9 0 0 4 2 0 0 0 0 7 10 0 0 8 6
,
 8 0 0 0 0 8 0 0 0 0 4 2 0 0 11 9
,
 12 12 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 12 12 0 0 0 0 1 0 0 0 9 12
`G:=sub<GL(4,GF(13))| [11,4,0,0,9,2,0,0,0,0,7,8,0,0,10,6],[8,0,0,0,0,8,0,0,0,0,4,11,0,0,2,9],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[1,12,0,0,0,12,0,0,0,0,1,9,0,0,0,12] >;`

C4⋊C4⋊S3 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4\rtimes S_3`
`% in TeX`

`G:=Group("C4:C4:S3");`
`// GroupNames label`

`G:=SmallGroup(96,105);`
`// by ID`

`G=gap.SmallGroup(96,105);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,55,218,188,86,2309]);`
`// Polycyclic`

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

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