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## G = C12.53D4order 96 = 25·3

### 10th non-split extension by C12 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.53D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C12.53D4
 Lower central C3 — C6 — C12 — C12.53D4
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for C12.53D4
G = < a,b,c | a12=1, b4=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a6b3 >

Character table of C12.53D4

 class 1 2A 2B 3 4A 4B 4C 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 24A 24B 24C 24D size 1 1 2 2 1 1 2 2 4 4 4 6 6 6 6 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 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 -i i -1 1 1 -1 i -i -1 -1 1 i i -i -i linear of order 4 ρ6 1 1 -1 1 -1 -1 1 1 -1 i -i 1 -1 -1 1 i -i -1 -1 1 -i -i i i linear of order 4 ρ7 1 1 -1 1 -1 -1 1 1 -1 i -i -1 1 1 -1 -i i -1 -1 1 -i -i i i linear of order 4 ρ8 1 1 -1 1 -1 -1 1 1 -1 -i i 1 -1 -1 1 -i i -1 -1 1 i i -i -i linear of order 4 ρ9 2 2 2 -1 2 2 2 -1 -1 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 -1 2 2 2 -1 -1 -2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ11 2 2 -2 2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 2 2 -1 -2 -2 -2 -1 -1 0 0 0 0 0 0 0 0 1 1 1 -√3 √3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ14 2 2 2 -1 -2 -2 -2 -1 -1 0 0 0 0 0 0 0 0 1 1 1 √3 -√3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ15 2 2 -2 -1 -2 -2 2 -1 1 2i -2i 0 0 0 0 0 0 1 1 -1 i i -i -i complex lifted from C4×S3 ρ16 2 2 -2 -1 -2 -2 2 -1 1 -2i 2i 0 0 0 0 0 0 1 1 -1 -i -i i i complex lifted from C4×S3 ρ17 2 2 -2 -1 2 2 -2 -1 1 0 0 0 0 0 0 0 0 -1 -1 1 -√-3 √-3 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 -2 -1 2 2 -2 -1 1 0 0 0 0 0 0 0 0 -1 -1 1 √-3 -√-3 -√-3 √-3 complex lifted from C3⋊D4 ρ19 2 -2 0 2 2i -2i 0 -2 0 0 0 -√-2 √2 -√2 √-2 0 0 2i -2i 0 0 0 0 0 complex lifted from C8.C4 ρ20 2 -2 0 2 -2i 2i 0 -2 0 0 0 -√-2 -√2 √2 √-2 0 0 -2i 2i 0 0 0 0 0 complex lifted from C8.C4 ρ21 2 -2 0 2 -2i 2i 0 -2 0 0 0 √-2 √2 -√2 -√-2 0 0 -2i 2i 0 0 0 0 0 complex lifted from C8.C4 ρ22 2 -2 0 2 2i -2i 0 -2 0 0 0 √-2 -√2 √2 -√-2 0 0 2i -2i 0 0 0 0 0 complex lifted from C8.C4 ρ23 4 -4 0 -2 4i -4i 0 2 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 0 complex faithful ρ24 4 -4 0 -2 -4i 4i 0 2 0 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 0 complex faithful

Smallest permutation representation of C12.53D4
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 25 4 28 7 31 10 34)(2 30 5 33 8 36 11 27)(3 35 6 26 9 29 12 32)(13 40 22 37 19 46 16 43)(14 45 23 42 20 39 17 48)(15 38 24 47 21 44 18 41)
(1 40 10 37 7 46 4 43)(2 45 11 42 8 39 5 48)(3 38 12 47 9 44 6 41)(13 34 22 31 19 28 16 25)(14 27 23 36 20 33 17 30)(15 32 24 29 21 26 18 35)```

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

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

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

Matrix representation of C12.53D4 in GL4(𝔽5) generated by

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

C12.53D4 in GAP, Magma, Sage, TeX

`C_{12}._{53}D_4`
`% in TeX`

`G:=Group("C12.53D4");`
`// GroupNames label`

`G:=SmallGroup(96,29);`
`// by ID`

`G=gap.SmallGroup(96,29);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,121,31,86,297,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^4=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^6*b^3>;`
`// generators/relations`

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