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## G = D12.6D4order 192 = 26·3

### 6th non-split extension by D12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.6D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — D12.C4 — D12.6D4
 Lower central C3 — C6 — C2×C12 — D12.6D4
 Upper central C1 — C2 — C2×C4 — C4.10D4

Generators and relations for D12.6D4
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=a6b, dbd-1=a3b, dcd-1=a3c3 >

Subgroups: 336 in 104 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], S3 [×2], C6, C6, C8 [×5], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, Dic3, C12 [×2], C12, D6 [×3], C2×C6, C2×C8 [×2], M4(2) [×2], M4(2) [×2], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8 [×2], C3⋊C8, C24 [×2], Dic6, C4×S3, D12, D12 [×2], C3⋊D4, C2×C12, C2×C12, C3×Q8 [×2], C22×S3, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, S3×C8, C8⋊S3, C24⋊C2, D24, C2×C3⋊C8, C4.Dic3, Q82S3 [×3], C3⋊Q16, C3×M4(2) [×2], C2×D12, C4○D12, C6×Q8, D4.3D4, C12.53D4, C12.46D4, C3×C4.10D4, D12.C4, C8⋊D6, C2×Q82S3, Q8.11D6, D12.6D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C22×S3, C4⋊D4, C4○D12, S3×D4 [×2], D4.3D4, Dic3⋊D4, D12.6D4

Character table of D12.6D4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 8E 8F 8G 12A 12B 12C 12D 24A 24B 24C 24D size 1 1 2 12 24 2 2 2 8 12 2 4 4 4 6 6 8 12 24 4 4 8 8 8 8 8 8 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 0 0 -1 2 2 2 0 -1 -1 2 2 0 0 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 0 0 2 2 -2 0 0 2 -2 0 0 2 2 0 -2 0 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 0 -1 2 2 -2 0 -1 -1 2 2 0 0 -2 0 0 -1 -1 1 1 1 -1 -1 1 orthogonal lifted from D6 ρ12 2 2 -2 -2 0 2 -2 2 0 2 2 -2 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 0 0 2 2 -2 0 0 2 -2 0 0 -2 -2 0 2 0 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 0 0 -1 2 2 2 0 -1 -1 -2 -2 0 0 -2 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ15 2 2 2 0 0 -1 2 2 -2 0 -1 -1 -2 -2 0 0 2 0 0 -1 -1 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ16 2 2 -2 2 0 2 -2 2 0 -2 2 -2 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 2 0 0 2 -2 -2 0 0 2 2 -2i 2i 0 0 0 0 0 -2 -2 0 0 0 2i -2i 0 complex lifted from C4○D4 ρ18 2 2 2 0 0 2 -2 -2 0 0 2 2 2i -2i 0 0 0 0 0 -2 -2 0 0 0 -2i 2i 0 complex lifted from C4○D4 ρ19 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 2i -2i 0 0 0 0 0 1 1 √-3 -√-3 -√3 i -i √3 complex lifted from C4○D12 ρ20 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 2i -2i 0 0 0 0 0 1 1 -√-3 √-3 √3 i -i -√3 complex lifted from C4○D12 ρ21 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -2i 2i 0 0 0 0 0 1 1 √-3 -√-3 √3 -i i -√3 complex lifted from C4○D12 ρ22 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -2i 2i 0 0 0 0 0 1 1 -√-3 √-3 -√3 -i i √3 complex lifted from C4○D12 ρ23 4 4 -4 0 0 -2 4 -4 0 0 -2 2 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 -4 0 0 -2 -4 4 0 0 -2 2 0 0 0 0 0 0 0 -2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 0 4 0 0 0 0 -4 0 0 0 2√-2 -2√-2 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.3D4 ρ26 4 -4 0 0 0 4 0 0 0 0 -4 0 0 0 -2√-2 2√-2 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.3D4 ρ27 8 -8 0 0 0 -4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of D12.6D4
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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)
(1 28 10 31 7 34 4 25)(2 35 11 26 8 29 5 32)(3 30 12 33 9 36 6 27)(13 46 22 37 19 40 16 43)(14 41 23 44 20 47 17 38)(15 48 24 39 21 42 18 45)
(1 46 4 37 7 40 10 43)(2 47 5 38 8 41 11 44)(3 48 6 39 9 42 12 45)(13 34 16 25 19 28 22 31)(14 35 17 26 20 29 23 32)(15 36 18 27 21 30 24 33)```

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

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

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

Matrix representation of D12.6D4 in GL6(𝔽73)

 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 71 0 0 0 0 1 72 0 0 0 0 67 12 0 72 0 0 6 0 1 0
,
 72 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 61 12 0 0 72 1 0 12 0 0 0 0 72 0 0 0 67 0 72 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 61 12 71 0 0 0 0 0 72 1 0 0 35 2 6 6 0 0 36 1 6 6
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 2 71 61 61 0 0 1 72 0 61 0 0 67 12 72 0 0 0 61 12 72 0

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,1,67,6,0,0,71,72,12,0,0,0,0,0,0,1,0,0,0,0,72,0],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,0,72,0,67,0,0,0,1,0,0,0,0,61,0,72,72,0,0,12,12,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,61,0,35,36,0,0,12,0,2,1,0,0,71,72,6,6,0,0,0,1,6,6],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,2,1,67,61,0,0,71,72,12,12,0,0,61,0,72,72,0,0,61,61,0,0] >;`

D12.6D4 in GAP, Magma, Sage, TeX

`D_{12}._6D_4`
`% in TeX`

`G:=Group("D12.6D4");`
`// GroupNames label`

`G:=SmallGroup(192,313);`
`// by ID`

`G=gap.SmallGroup(192,313);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,219,184,297,136,1684,851,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^3>;`
`// generators/relations`

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