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## G = C6.5S4order 144 = 24·32

### 5th non-split extension by C6 of S4 acting via S4/A4=C2

Aliases: C6.5S4, C3⋊CSU2(𝔽3), SL2(𝔽3).S3, Q8.(C3⋊S3), C2.2(C3⋊S4), (C3×Q8).3S3, (C3×SL2(𝔽3)).1C2, SmallGroup(144,124)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — C6.5S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C6.5S4
 Lower central C3×SL2(𝔽3) — C6.5S4
 Upper central C1 — C2

Generators and relations for C6.5S4
G = < a,b,c,d,e | a6=d3=1, b2=c2=e2=a3, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a3b, dbd-1=a3bc, ebe-1=bc, dcd-1=b, ece-1=a3c, ede-1=d-1 >

Character table of C6.5S4

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 8A 8B 12 size 1 1 2 8 8 8 6 36 2 8 8 8 18 18 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 -1 -1 2 -1 2 0 -1 2 -1 -1 0 0 -1 orthogonal lifted from S3 ρ4 2 2 2 -1 -1 -1 2 0 2 -1 -1 -1 0 0 2 orthogonal lifted from S3 ρ5 2 2 -1 2 -1 -1 2 0 -1 -1 2 -1 0 0 -1 orthogonal lifted from S3 ρ6 2 2 -1 -1 -1 2 2 0 -1 -1 -1 2 0 0 -1 orthogonal lifted from S3 ρ7 2 -2 2 -1 -1 -1 0 0 -2 1 1 1 -√2 √2 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ8 2 -2 2 -1 -1 -1 0 0 -2 1 1 1 √2 -√2 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ9 3 3 3 0 0 0 -1 1 3 0 0 0 -1 -1 -1 orthogonal lifted from S4 ρ10 3 3 3 0 0 0 -1 -1 3 0 0 0 1 1 -1 orthogonal lifted from S4 ρ11 4 -4 -2 1 -2 1 0 0 2 2 -1 -1 0 0 0 symplectic faithful, Schur index 2 ρ12 4 -4 4 1 1 1 0 0 -4 -1 -1 -1 0 0 0 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ13 4 -4 -2 1 1 -2 0 0 2 -1 -1 2 0 0 0 symplectic faithful, Schur index 2 ρ14 4 -4 -2 -2 1 1 0 0 2 -1 2 -1 0 0 0 symplectic faithful, Schur index 2 ρ15 6 6 -3 0 0 0 -2 0 -3 0 0 0 0 0 1 orthogonal lifted from C3⋊S4

Smallest permutation representation of C6.5S4
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 22 4 19)(2 23 5 20)(3 24 6 21)(7 37 10 40)(8 38 11 41)(9 39 12 42)(13 43 16 46)(14 44 17 47)(15 45 18 48)(25 31 28 34)(26 32 29 35)(27 33 30 36)
(1 33 4 36)(2 34 5 31)(3 35 6 32)(7 14 10 17)(8 15 11 18)(9 16 12 13)(19 30 22 27)(20 25 23 28)(21 26 24 29)(37 47 40 44)(38 48 41 45)(39 43 42 46)
(7 37 47)(8 38 48)(9 39 43)(10 40 44)(11 41 45)(12 42 46)(19 36 27)(20 31 28)(21 32 29)(22 33 30)(23 34 25)(24 35 26)
(1 13 4 16)(2 18 5 15)(3 17 6 14)(7 35 10 32)(8 34 11 31)(9 33 12 36)(19 39 22 42)(20 38 23 41)(21 37 24 40)(25 45 28 48)(26 44 29 47)(27 43 30 46)```

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

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

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

C6.5S4 is a maximal subgroup of
CSU2(𝔽3)⋊S3  Dic3.4S4  S3×CSU2(𝔽3)  D6.S4  SL2(𝔽3).D6  C12.6S4  C12.14S4  C32⋊CSU2(𝔽3)  C18.5S4  C324CSU2(𝔽3)
C6.5S4 is a maximal quotient of
C6.GL2(𝔽3)  C18.5S4  C32.3CSU2(𝔽3)  C322CSU2(𝔽3)  C324CSU2(𝔽3)

Matrix representation of C6.5S4 in GL4(𝔽5) generated by

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

C6.5S4 in GAP, Magma, Sage, TeX

`C_6._5S_4`
`% in TeX`

`G:=Group("C6.5S4");`
`// GroupNames label`

`G:=SmallGroup(144,124);`
`// by ID`

`G=gap.SmallGroup(144,124);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-2,2,-2,432,49,218,867,1305,117,544,820,202,88]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=d^3=1,b^2=c^2=e^2=a^3,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^3*b,d*b*d^-1=a^3*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^3*c,e*d*e^-1=d^-1>;`
`// generators/relations`

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