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G = C12.47D4order 96 = 25·3

4th non-split extension by C12 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C12.47D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12.47D4
 Lower central C3 — C6 — C2×C6 — C12.47D4
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C12.47D4
G = < a,b,c | a12=1, b4=c2=a6, bab-1=cac-1=a-1, cbc-1=a9b3 >

Character table of C12.47D4

 class 1 2A 2B 3 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 12A 12B 12C 24A 24B 24C 24D size 1 1 2 2 2 2 12 12 2 4 4 4 12 12 2 2 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 trivial ρ2 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 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 linear of order 2 ρ5 1 1 1 1 -1 -1 1 -1 1 1 -i i i -i -1 -1 -1 -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 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 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 -i -i i i linear of order 4 ρ9 2 2 -2 2 -2 2 0 0 2 -2 0 0 0 0 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -1 2 2 0 0 -1 -1 -2 -2 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ11 2 2 2 -1 2 2 0 0 -1 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 -2 2 2 -2 0 0 2 -2 0 0 0 0 2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 -1 -2 2 0 0 -1 1 0 0 0 0 1 1 -1 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ14 2 2 -2 -1 -2 2 0 0 -1 1 0 0 0 0 1 1 -1 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ15 2 2 2 -1 -2 -2 0 0 -1 -1 2i -2i 0 0 1 1 1 -i -i i i complex lifted from C4×S3 ρ16 2 2 2 -1 -2 -2 0 0 -1 -1 -2i 2i 0 0 1 1 1 i i -i -i complex lifted from C4×S3 ρ17 2 2 -2 -1 2 -2 0 0 -1 1 0 0 0 0 -1 -1 1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 -2 -1 2 -2 0 0 -1 1 0 0 0 0 -1 -1 1 √-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ19 4 -4 0 4 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2 ρ20 4 -4 0 -2 0 0 0 0 2 0 0 0 0 0 -2√3 2√3 0 0 0 0 0 symplectic faithful, Schur index 2 ρ21 4 -4 0 -2 0 0 0 0 2 0 0 0 0 0 2√3 -2√3 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C12.47D4
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 43 10 46 7 37 4 40)(2 42 11 45 8 48 5 39)(3 41 12 44 9 47 6 38)(13 26 16 35 19 32 22 29)(14 25 17 34 20 31 23 28)(15 36 18 33 21 30 24 27)
(1 25 7 31)(2 36 8 30)(3 35 9 29)(4 34 10 28)(5 33 11 27)(6 32 12 26)(13 47 19 41)(14 46 20 40)(15 45 21 39)(16 44 22 38)(17 43 23 37)(18 42 24 48)```

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

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

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

Matrix representation of C12.47D4 in GL4(𝔽73) generated by

 66 66 0 0 7 59 0 0 0 0 66 66 0 0 7 59
,
 0 0 71 20 0 0 18 2 47 34 0 0 8 26 0 0
,
 71 20 0 0 18 2 0 0 0 0 71 20 0 0 18 2
`G:=sub<GL(4,GF(73))| [66,7,0,0,66,59,0,0,0,0,66,7,0,0,66,59],[0,0,47,8,0,0,34,26,71,18,0,0,20,2,0,0],[71,18,0,0,20,2,0,0,0,0,71,18,0,0,20,2] >;`

C12.47D4 in GAP, Magma, Sage, TeX

`C_{12}._{47}D_4`
`% in TeX`

`G:=Group("C12.47D4");`
`// GroupNames label`

`G:=SmallGroup(96,31);`
`// by ID`

`G=gap.SmallGroup(96,31);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,121,31,362,86,297,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^4=c^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^3>;`
`// generators/relations`

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