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## G = C12.48D4order 96 = 25·3

### 5th non-split extension by C12 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C12.48D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C2×Dic6 — C12.48D4
 Lower central C3 — C2×C6 — C12.48D4
 Upper central C1 — C22 — C22×C4

Generators and relations for C12.48D4
G = < a,b,c | a12=b4=1, c2=a6, bab-1=cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 146 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, Dic3 [×4], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic6 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×C6, C22⋊Q8, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C2×Dic6, C22×C12, C12.48D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, C2×Dic6, C4○D12, C2×C3⋊D4, C12.48D4

Character table of C12.48D4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 2 2 2 2 2 2 2 12 12 12 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 2 2 2 2 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 -2 -2 -1 -2 -2 2 2 0 0 0 0 -1 -1 1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 orthogonal lifted from D6 ρ11 2 2 -2 -2 0 0 2 2 -2 0 0 0 0 0 0 -2 -2 0 0 2 0 0 0 -2 0 0 -2 2 0 2 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 -1 2 2 -2 -2 0 0 0 0 -1 -1 1 1 -1 1 1 1 -1 1 1 -1 -1 1 -1 orthogonal lifted from D6 ρ13 2 2 -2 -2 0 0 2 -2 2 0 0 0 0 0 0 -2 -2 0 0 2 0 0 0 2 0 0 2 -2 0 -2 orthogonal lifted from D4 ρ14 2 2 2 2 2 2 -1 -2 -2 -2 -2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 orthogonal lifted from D6 ρ15 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 -2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 -2 2 -2 2 -2 -1 0 0 0 0 0 0 0 0 1 -1 -1 -1 1 1 1 -√3 √3 -√3 √3 -√3 -√3 √3 √3 symplectic lifted from Dic6, Schur index 2 ρ18 2 -2 2 -2 -2 2 -1 0 0 0 0 0 0 0 0 1 -1 1 1 1 -1 -1 -√3 -√3 -√3 √3 √3 √3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ19 2 -2 2 -2 2 -2 -1 0 0 0 0 0 0 0 0 1 -1 -1 -1 1 1 1 √3 -√3 √3 -√3 √3 √3 -√3 -√3 symplectic lifted from Dic6, Schur index 2 ρ20 2 -2 2 -2 -2 2 -1 0 0 0 0 0 0 0 0 1 -1 1 1 1 -1 -1 √3 √3 √3 -√3 -√3 -√3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ21 2 2 -2 -2 0 0 -1 -2 2 0 0 0 0 0 0 1 1 √-3 -√-3 -1 √-3 -√-3 √-3 -1 -√-3 √-3 -1 1 -√-3 1 complex lifted from C3⋊D4 ρ22 2 2 -2 -2 0 0 -1 2 -2 0 0 0 0 0 0 1 1 -√-3 √-3 -1 -√-3 √-3 √-3 1 -√-3 √-3 1 -1 -√-3 -1 complex lifted from C3⋊D4 ρ23 2 2 -2 -2 0 0 -1 -2 2 0 0 0 0 0 0 1 1 -√-3 √-3 -1 -√-3 √-3 -√-3 -1 √-3 -√-3 -1 1 √-3 1 complex lifted from C3⋊D4 ρ24 2 2 -2 -2 0 0 -1 2 -2 0 0 0 0 0 0 1 1 √-3 -√-3 -1 √-3 -√-3 -√-3 1 √-3 -√-3 1 -1 √-3 -1 complex lifted from C3⋊D4 ρ25 2 -2 -2 2 0 0 2 0 0 -2i 2i 0 0 0 0 2 -2 0 0 -2 0 0 -2i 0 2i 2i 0 0 -2i 0 complex lifted from C4○D4 ρ26 2 -2 -2 2 0 0 2 0 0 2i -2i 0 0 0 0 2 -2 0 0 -2 0 0 2i 0 -2i -2i 0 0 2i 0 complex lifted from C4○D4 ρ27 2 -2 -2 2 0 0 -1 0 0 -2i 2i 0 0 0 0 -1 1 √-3 -√-3 1 -√-3 √-3 i -√3 -i -i √3 -√3 i √3 complex lifted from C4○D12 ρ28 2 -2 -2 2 0 0 -1 0 0 -2i 2i 0 0 0 0 -1 1 -√-3 √-3 1 √-3 -√-3 i √3 -i -i -√3 √3 i -√3 complex lifted from C4○D12 ρ29 2 -2 -2 2 0 0 -1 0 0 2i -2i 0 0 0 0 -1 1 √-3 -√-3 1 -√-3 √-3 -i √3 i i -√3 √3 -i -√3 complex lifted from C4○D12 ρ30 2 -2 -2 2 0 0 -1 0 0 2i -2i 0 0 0 0 -1 1 -√-3 √-3 1 √-3 -√-3 -i -√3 i i √3 -√3 -i √3 complex lifted from C4○D12

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

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

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

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

Matrix representation of C12.48D4 in GL4(𝔽13) generated by

 6 0 0 0 0 11 0 0 0 0 5 5 0 0 0 8
,
 0 1 0 0 1 0 0 0 0 0 10 7 0 0 6 3
,
 0 1 0 0 12 0 0 0 0 0 10 7 0 0 6 3
`G:=sub<GL(4,GF(13))| [6,0,0,0,0,11,0,0,0,0,5,0,0,0,5,8],[0,1,0,0,1,0,0,0,0,0,10,6,0,0,7,3],[0,12,0,0,1,0,0,0,0,0,10,6,0,0,7,3] >;`

C12.48D4 in GAP, Magma, Sage, TeX

`C_{12}._{48}D_4`
`% in TeX`

`G:=Group("C12.48D4");`
`// GroupNames label`

`G:=SmallGroup(96,131);`
`// by ID`

`G=gap.SmallGroup(96,131);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,217,103,218,2309]);`
`// Polycyclic`

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

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