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## G = D12.2D4order 192 = 26·3

### 2nd non-split extension by D12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.2D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — D12.C4 — D12.2D4
 Lower central C3 — C6 — C2×C12 — D12.2D4
 Upper central C1 — C2 — C2×C4 — C4.D4

Generators and relations for D12.2D4
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=a6b, dbd-1=a3b, dcd-1=a3c3 >

Subgroups: 304 in 104 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], S3, C6, C6 [×2], C8 [×5], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, Dic3 [×2], C12 [×2], D6, C2×C6, C2×C6 [×2], 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, Dic6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4 [×2], C22×C6, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, S3×C8, C8⋊S3, C24⋊C2, Dic12, C2×C3⋊C8, C4.Dic3, D4⋊S3, D4.S3 [×3], C3×M4(2) [×2], C2×Dic6, C4○D12, C6×D4, D4.3D4, C12.53D4, C12.47D4, C3×C4.D4, D12.C4, C8.D6, D126C22, C2×D4.S3, D12.2D4
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, D12.2D4

Character table of D12.2D4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 8C 8D 8E 8F 8G 12A 12B 24A 24B 24C 24D size 1 1 2 8 12 2 2 2 12 24 2 4 8 8 4 4 6 6 8 12 24 4 4 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 -2 0 -1 2 2 0 0 -1 -1 1 1 2 2 0 0 -2 0 0 -1 -1 1 -1 -1 1 orthogonal lifted from D6 ρ10 2 2 -2 0 2 2 -2 2 -2 0 2 -2 0 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 -1 2 2 0 0 -1 -1 -1 -1 -2 -2 0 0 -2 0 0 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ12 2 2 -2 0 -2 2 -2 2 2 0 2 -2 0 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 -1 2 2 0 0 -1 -1 -1 -1 2 2 0 0 2 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 -2 0 0 2 2 -2 0 0 2 -2 0 0 0 0 -2 -2 0 2 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 -2 0 -1 2 2 0 0 -1 -1 1 1 -2 -2 0 0 2 0 0 -1 -1 -1 1 1 -1 orthogonal lifted from D6 ρ16 2 2 -2 0 0 2 2 -2 0 0 2 -2 0 0 0 0 2 2 0 -2 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 2 0 0 2 -2 -2 0 0 2 2 0 0 2i -2i 0 0 0 0 0 -2 -2 0 2i -2i 0 complex lifted from C4○D4 ρ18 2 2 2 0 0 2 -2 -2 0 0 2 2 0 0 -2i 2i 0 0 0 0 0 -2 -2 0 -2i 2i 0 complex lifted from C4○D4 ρ19 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 √-3 -√-3 -2i 2i 0 0 0 0 0 1 1 -√3 i -i √3 complex lifted from C4○D12 ρ20 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -√-3 √-3 -2i 2i 0 0 0 0 0 1 1 √3 i -i -√3 complex lifted from C4○D12 ρ21 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -√-3 √-3 2i -2i 0 0 0 0 0 1 1 -√3 -i i √3 complex lifted from C4○D12 ρ22 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 √-3 -√-3 2i -2i 0 0 0 0 0 1 1 √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 0 0 -2 2 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 0 0 2 -2 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 0 4 0 0 0 0 -4 0 0 0 0 0 2√-2 -2√-2 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 0 0 -2√-2 2√-2 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 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of D12.2D4 in GL6(𝔽73)

 64 0 0 0 0 0 71 8 0 0 0 0 0 0 72 29 0 0 0 0 10 1 0 0 0 0 22 45 1 63 0 0 45 29 44 72
,
 8 5 0 0 0 0 2 65 0 0 0 0 0 0 32 45 56 12 0 0 59 0 12 0 0 0 13 10 41 14 0 0 10 28 28 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 27 0 29 0 0 0 0 1 0 10 0 0 0 1 46 0 0 0 1 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 27 0 29 0 0 0 22 72 2 63 0 0 0 1 46 0 0 0 27 44 29 1

`G:=sub<GL(6,GF(73))| [64,71,0,0,0,0,0,8,0,0,0,0,0,0,72,10,22,45,0,0,29,1,45,29,0,0,0,0,1,44,0,0,0,0,63,72],[8,2,0,0,0,0,5,65,0,0,0,0,0,0,32,59,13,10,0,0,45,0,10,28,0,0,56,12,41,28,0,0,12,0,14,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,1,0,0,0,1,1,0,0,0,29,0,46,0,0,0,0,10,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,22,0,27,0,0,0,72,1,44,0,0,29,2,46,29,0,0,0,63,0,1] >;`

D12.2D4 in GAP, Magma, Sage, TeX

`D_{12}._2D_4`
`% in TeX`

`G:=Group("D12.2D4");`
`// GroupNames label`

`G:=SmallGroup(192,307);`
`// by ID`

`G=gap.SmallGroup(192,307);`
`# by ID`

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

`G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^9,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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