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G = D12.4D4order 192 = 26·3

4th non-split extension by D12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.4D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — Q8.15D6 — D12.4D4
 Lower central C3 — C6 — C2×C12 — D12.4D4
 Upper central C1 — C2 — C2×C4 — C4.10D4

Generators and relations for D12.4D4
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, dbd-1=a9b, dcd-1=a3c3 >

Subgroups: 432 in 142 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C24⋊C2, Dic12, C4×Dic3, Dic3⋊C4, C3×M4(2), C2×Dic6, C4○D12, C4○D12, S3×Q8, Q83S3, C6×Q8, D4.10D4, D12⋊C4, C3×C4.10D4, C8.D6, Dic3⋊Q8, Q8.15D6, D12.4D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, D4.10D4, D6⋊D4, D12.4D4

Character table of D12.4D4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 8A 8B 12A 12B 12C 12D 24A 24B 24C 24D size 1 1 2 12 12 2 2 2 4 4 12 12 12 12 24 2 4 8 8 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 -2 0 0 0 0 0 -1 -1 -2 2 -1 -1 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ10 2 2 2 0 0 -1 2 2 2 2 0 0 0 0 0 -1 -1 2 2 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 0 0 2 -2 -2 -2 2 0 0 0 0 0 2 2 0 0 -2 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 -2 2 -2 2 0 0 0 2 0 0 0 2 -2 0 0 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 0 0 2 -2 -2 2 -2 0 0 0 0 0 2 2 0 0 -2 -2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 2 2 -2 2 0 0 0 -2 0 0 0 2 -2 0 0 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 0 0 -1 2 2 -2 -2 0 0 0 0 0 -1 -1 2 -2 -1 -1 1 1 1 -1 -1 1 orthogonal lifted from D6 ρ16 2 2 2 0 0 -1 2 2 2 2 0 0 0 0 0 -1 -1 -2 -2 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ17 2 2 -2 -2 0 2 2 -2 0 0 0 0 2 0 0 2 -2 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 2 0 2 2 -2 0 0 0 0 -2 0 0 2 -2 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 0 0 -1 -2 -2 2 -2 0 0 0 0 0 -1 -1 0 0 1 1 -1 1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ20 2 2 2 0 0 -1 -2 -2 2 -2 0 0 0 0 0 -1 -1 0 0 1 1 -1 1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ21 2 2 2 0 0 -1 -2 -2 -2 2 0 0 0 0 0 -1 -1 0 0 1 1 1 -1 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ22 2 2 2 0 0 -1 -2 -2 -2 2 0 0 0 0 0 -1 -1 0 0 1 1 1 -1 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ23 4 4 -4 0 0 -2 -4 4 0 0 0 0 0 0 0 -2 2 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 0 0 0 0 0 -2 2 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 2 0 0 -2 0 -4 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ26 4 -4 0 0 0 4 0 0 0 0 -2 0 0 2 0 -4 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ27 8 -8 0 0 0 -4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of D12.4D4 in GL8(𝔽73)

 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 1 71 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 72 2 0 0 0 0 0 0 72 1
,
 66 0 7 0 0 0 0 0 0 66 0 7 0 0 0 0 14 0 7 0 0 0 0 0 0 14 0 7 0 0 0 0 0 0 0 0 0 0 1 71 0 0 0 0 0 0 1 72 0 0 0 0 72 2 0 0 0 0 0 0 72 1 0 0
,
 0 66 0 59 0 0 0 0 7 0 14 0 0 0 0 0 0 14 0 7 0 0 0 0 59 0 66 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 71 0 0 0 0 0 0 1 72 0 0
,
 66 0 59 0 0 0 0 0 0 7 0 14 0 0 0 0 14 0 7 0 0 0 0 0 0 59 0 66 0 0 0 0 0 0 0 0 0 0 51 17 0 0 0 0 0 0 23 22 0 0 0 0 68 27 0 0 0 0 0 0 45 5 0 0

`G:=sub<GL(8,GF(73))| [1,0,72,0,0,0,0,0,0,1,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1],[66,0,14,0,0,0,0,0,0,66,0,14,0,0,0,0,7,0,7,0,0,0,0,0,0,7,0,7,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0],[0,7,0,59,0,0,0,0,66,0,14,0,0,0,0,0,0,14,0,66,0,0,0,0,59,0,7,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,71,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[66,0,14,0,0,0,0,0,0,7,0,59,0,0,0,0,59,0,7,0,0,0,0,0,0,14,0,66,0,0,0,0,0,0,0,0,0,0,68,45,0,0,0,0,0,0,27,5,0,0,0,0,51,23,0,0,0,0,0,0,17,22,0,0] >;`

D12.4D4 in GAP, Magma, Sage, TeX

`D_{12}._4D_4`
`% in TeX`

`G:=Group("D12.4D4");`
`// GroupNames label`

`G:=SmallGroup(192,311);`
`// by ID`

`G=gap.SmallGroup(192,311);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,58,1123,570,136,1684,438,6278]);`
`// 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^3*b,d*b*d^-1=a^9*b,d*c*d^-1=a^3*c^3>;`
`// generators/relations`

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