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## G = C3⋊S3.Q8order 144 = 24·32

### The non-split extension by C3⋊S3 of Q8 acting via Q8/C2=C22

Aliases: C3⋊S3.Q8, C2.2S3≀C2, C32⋊C41C4, (C3×C6).2D4, C321(C4⋊C4), C6.D6.2C2, C3⋊S3.3(C2×C4), (C2×C32⋊C4).1C2, (C2×C3⋊S3).2C22, SmallGroup(144,116)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C3⋊S3.Q8
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C6.D6 — C3⋊S3.Q8
 Lower central C32 — C3⋊S3 — C3⋊S3.Q8
 Upper central C1 — C2

Generators and relations for C3⋊S3.Q8
G = < a,b,c,d,e | a3=b3=c2=d4=1, e2=d2, ab=ba, cac=dbd-1=a-1, dad-1=cbc=ebe-1=b-1, ae=ea, cd=dc, ce=ec, ede-1=cd-1 >

Character table of C3⋊S3.Q8

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 12A 12B 12C 12D size 1 1 9 9 4 4 6 6 6 6 18 18 4 4 12 12 12 12 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 -1 1 1 1 -i i i -i -1 1 -1 -1 i -i i -i linear of order 4 ρ6 1 -1 -1 1 1 1 i i -i -i 1 -1 -1 -1 i i -i -i linear of order 4 ρ7 1 -1 -1 1 1 1 -i -i i i 1 -1 -1 -1 -i -i i i linear of order 4 ρ8 1 -1 -1 1 1 1 i -i -i i -1 1 -1 -1 -i i -i i linear of order 4 ρ9 2 2 -2 -2 2 2 0 0 0 0 0 0 2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 4 4 0 0 1 -2 0 2 0 2 0 0 -2 1 -1 0 0 -1 orthogonal lifted from S3≀C2 ρ12 4 4 0 0 -2 1 2 0 2 0 0 0 1 -2 0 -1 -1 0 orthogonal lifted from S3≀C2 ρ13 4 4 0 0 -2 1 -2 0 -2 0 0 0 1 -2 0 1 1 0 orthogonal lifted from S3≀C2 ρ14 4 4 0 0 1 -2 0 -2 0 -2 0 0 -2 1 1 0 0 1 orthogonal lifted from S3≀C2 ρ15 4 -4 0 0 -2 1 -2i 0 2i 0 0 0 -1 2 0 i -i 0 complex faithful ρ16 4 -4 0 0 1 -2 0 -2i 0 2i 0 0 2 -1 i 0 0 -i complex faithful ρ17 4 -4 0 0 1 -2 0 2i 0 -2i 0 0 2 -1 -i 0 0 i complex faithful ρ18 4 -4 0 0 -2 1 2i 0 -2i 0 0 0 -1 2 0 -i i 0 complex faithful

Permutation representations of C3⋊S3.Q8
On 24 points - transitive group 24T214
Generators in S24
```(1 20 15)(3 18 13)(6 11 24)(8 9 22)
(2 16 17)(4 14 19)(5 23 10)(7 21 12)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 20)(14 17)(15 18)(16 19)(21 23)(22 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 24 3 22)(2 21 4 23)(5 16 7 14)(6 18 8 20)(9 15 11 13)(10 17 12 19)```

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

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

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

`G:=TransitiveGroup(24,214);`

On 24 points - transitive group 24T216
Generators in S24
```(1 13 18)(3 15 20)(6 11 22)(8 9 24)
(2 19 14)(4 17 16)(5 21 10)(7 23 12)
(5 10)(6 11)(7 12)(8 9)(13 18)(14 19)(15 20)(16 17)
(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 24 3 22)(2 23 4 21)(5 19 7 17)(6 13 8 15)(9 20 11 18)(10 14 12 16)```

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

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

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

`G:=TransitiveGroup(24,216);`

C3⋊S3.Q8 is a maximal subgroup of
PSU3(𝔽2)⋊C4  F9⋊C4  S32⋊Q8  C32⋊C4⋊Q8  C4×S3≀C2  C62.9D4  C62⋊D4  C33⋊C4⋊C4  (C3×C6).9D12
C3⋊S3.Q8 is a maximal quotient of
C32⋊C4⋊C8  C4.19S3≀C2  C62.D4  C62.6D4  C62.7D4  C6.S3≀C2  C33⋊C4⋊C4  (C3×C6).9D12

Matrix representation of C3⋊S3.Q8 in GL4(𝔽5) generated by

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

C3⋊S3.Q8 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3.Q_8`
`% in TeX`

`G:=Group("C3:S3.Q8");`
`// GroupNames label`

`G:=SmallGroup(144,116);`
`// by ID`

`G=gap.SmallGroup(144,116);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,3,48,73,55,964,730,256,299,881]);`
`// Polycyclic`

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

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