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## G = C12.31D6order 144 = 24·32

### 5th non-split extension by C12 of D6 acting via D6/S3=C2

Aliases: C12.31D6, C323M4(2), C3⋊C85S3, C4.16S32, C6.2(C4×S3), C31(C8⋊S3), C3⋊Dic3.3C4, (C3×C12).30C22, C2.3(C6.D6), (C3×C3⋊C8)⋊8C2, (C2×C3⋊S3).3C4, (C4×C3⋊S3).4C2, (C3×C6).11(C2×C4), SmallGroup(144,55)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12.31D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C3⋊C8 — C12.31D6
 Lower central C32 — C3×C6 — C12.31D6
 Upper central C1 — C4

Generators and relations for C12.31D6
G = < a,b,c | a12=c2=1, b6=a3, bab-1=cac=a5, cbc=b5 >

Character table of C12.31D6

 class 1 2A 2B 3A 3B 3C 4A 4B 4C 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 18 2 2 4 1 1 18 2 2 4 6 6 6 6 2 2 2 2 4 4 6 6 6 6 6 6 6 6 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 -i i i -i -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 1 1 -i -i i i -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 1 1 i i -i -i -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 1 1 i -i -i i -1 -1 -1 -1 -1 -1 -i i i -i -i -i i i linear of order 4 ρ9 2 2 0 2 -1 -1 2 2 0 -1 2 -1 2 0 2 0 2 -1 2 -1 -1 -1 -1 0 0 -1 0 0 -1 -1 orthogonal lifted from S3 ρ10 2 2 0 -1 2 -1 2 2 0 2 -1 -1 0 -2 0 -2 -1 2 -1 2 -1 -1 0 1 1 0 1 1 0 0 orthogonal lifted from D6 ρ11 2 2 0 -1 2 -1 2 2 0 2 -1 -1 0 2 0 2 -1 2 -1 2 -1 -1 0 -1 -1 0 -1 -1 0 0 orthogonal lifted from S3 ρ12 2 2 0 2 -1 -1 2 2 0 -1 2 -1 -2 0 -2 0 2 -1 2 -1 -1 -1 1 0 0 1 0 0 1 1 orthogonal lifted from D6 ρ13 2 -2 0 2 2 2 2i -2i 0 -2 -2 -2 0 0 0 0 2i -2i -2i 2i -2i 2i 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ14 2 2 0 2 -1 -1 -2 -2 0 -1 2 -1 2i 0 -2i 0 -2 1 -2 1 1 1 i 0 0 i 0 0 -i -i complex lifted from C4×S3 ρ15 2 2 0 -1 2 -1 -2 -2 0 2 -1 -1 0 -2i 0 2i 1 -2 1 -2 1 1 0 -i -i 0 i i 0 0 complex lifted from C4×S3 ρ16 2 2 0 2 -1 -1 -2 -2 0 -1 2 -1 -2i 0 2i 0 -2 1 -2 1 1 1 -i 0 0 -i 0 0 i i complex lifted from C4×S3 ρ17 2 2 0 -1 2 -1 -2 -2 0 2 -1 -1 0 2i 0 -2i 1 -2 1 -2 1 1 0 i i 0 -i -i 0 0 complex lifted from C4×S3 ρ18 2 -2 0 2 2 2 -2i 2i 0 -2 -2 -2 0 0 0 0 -2i 2i 2i -2i 2i -2i 0 0 0 0 0 0 0 0 complex lifted from M4(2) ρ19 2 -2 0 -1 2 -1 -2i 2i 0 -2 1 1 0 0 0 0 i 2i -i -2i -i i 0 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 0 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 0 0 complex lifted from C8⋊S3 ρ20 2 -2 0 -1 2 -1 2i -2i 0 -2 1 1 0 0 0 0 -i -2i i 2i i -i 0 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 0 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 0 0 complex lifted from C8⋊S3 ρ21 2 -2 0 -1 2 -1 -2i 2i 0 -2 1 1 0 0 0 0 i 2i -i -2i -i i 0 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 0 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 0 0 complex lifted from C8⋊S3 ρ22 2 -2 0 -1 2 -1 2i -2i 0 -2 1 1 0 0 0 0 -i -2i i 2i i -i 0 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 0 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 0 0 complex lifted from C8⋊S3 ρ23 2 -2 0 2 -1 -1 -2i 2i 0 1 -2 1 0 0 0 0 -2i -i 2i i -i i 2ζ8ζ3+ζ8 0 0 2ζ85ζ3+ζ85 0 0 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 complex lifted from C8⋊S3 ρ24 2 -2 0 2 -1 -1 -2i 2i 0 1 -2 1 0 0 0 0 -2i -i 2i i -i i 2ζ85ζ3+ζ85 0 0 2ζ8ζ3+ζ8 0 0 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 complex lifted from C8⋊S3 ρ25 2 -2 0 2 -1 -1 2i -2i 0 1 -2 1 0 0 0 0 2i i -2i -i i -i 2ζ87ζ3+ζ87 0 0 2ζ83ζ3+ζ83 0 0 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 complex lifted from C8⋊S3 ρ26 2 -2 0 2 -1 -1 2i -2i 0 1 -2 1 0 0 0 0 2i i -2i -i i -i 2ζ83ζ3+ζ83 0 0 2ζ87ζ3+ζ87 0 0 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 complex lifted from C8⋊S3 ρ27 4 4 0 -2 -2 1 -4 -4 0 -2 -2 1 0 0 0 0 2 2 2 2 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C6.D6 ρ28 4 4 0 -2 -2 1 4 4 0 -2 -2 1 0 0 0 0 -2 -2 -2 -2 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ29 4 -4 0 -2 -2 1 4i -4i 0 2 2 -1 0 0 0 0 -2i 2i 2i -2i -i i 0 0 0 0 0 0 0 0 complex faithful ρ30 4 -4 0 -2 -2 1 -4i 4i 0 2 2 -1 0 0 0 0 2i -2i -2i 2i i -i 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C12.31D6
On 24 points - transitive group 24T236
Generators in S24
```(1 3 5 7 9 11 13 15 17 19 21 23)(2 12 22 8 18 4 14 24 10 20 6 16)
(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 21)(3 7)(4 12)(5 17)(6 22)(9 13)(10 18)(11 23)(15 19)(16 24)```

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

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

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

`G:=TransitiveGroup(24,236);`

C12.31D6 is a maximal subgroup of
C4.S3≀C2  (C3×C12).D4  C32⋊C4≀C2  C4.19S3≀C2  S3×C8⋊S3  C24.63D6  C24.D6  C3⋊C8.22D6  C3⋊C820D6  D12.D6  Dic6.D6  D125D6  D12.10D6  Dic6.10D6  D12.15D6  C36.40D6  He3⋊M4(2)  C339M4(2)  C3310M4(2)
C12.31D6 is a maximal quotient of
C2.Dic32  C12.78D12  C12.15Dic6  C36.40D6  He33M4(2)  C339M4(2)  C3310M4(2)

Matrix representation of C12.31D6 in GL4(𝔽5) generated by

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

C12.31D6 in GAP, Magma, Sage, TeX

`C_{12}._{31}D_6`
`% in TeX`

`G:=Group("C12.31D6");`
`// GroupNames label`

`G:=SmallGroup(144,55);`
`// by ID`

`G=gap.SmallGroup(144,55);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,31,50,490,3461]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=c^2=1,b^6=a^3,b*a*b^-1=c*a*c=a^5,c*b*c=b^5>;`
`// generators/relations`

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