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## G = D12.40D4order 192 = 26·3

### 10th non-split extension by D12 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.40D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — Q8○D12 — D12.40D4
 Lower central C3 — C6 — C2×C12 — D12.40D4
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for D12.40D4
G = < a,b,c,d | a12=b2=1, c4=d2=a6, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=a6c3 >

Subgroups: 400 in 142 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C4⋊C4, M4(2), M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, C8.C22, 2- 1+4, C4.Dic3, C4×Dic3, Dic3⋊C4, Q82S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C2×Dic6, C2×Dic6, C4○D12, C4○D12, D42S3, S3×Q8, C6×Q8, C3×C4○D4, D4.10D4, C12.47D4, D12⋊C4, Q83Dic3, Q8.11D6, Dic3⋊Q8, C3×C8.C22, Q8○D12, D12.40D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, D4.10D4, C232D6, D12.40D4

Character table of D12.40D4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 24A 24B size 1 1 2 4 12 2 2 2 4 8 12 12 12 12 12 2 4 8 8 24 4 4 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 2 2 -2 2 0 0 0 -2 0 0 0 2 -2 0 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 0 2 2 -2 2 0 0 0 0 0 0 2 -2 -2 0 0 -2 2 0 0 2 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 -1 2 2 2 2 0 0 0 0 0 -1 -1 -1 2 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 -2 0 -2 2 -2 2 0 0 0 2 0 0 0 2 -2 0 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 -2 0 -1 2 2 -2 2 0 0 0 0 0 -1 -1 1 -2 0 -1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ14 2 2 -2 2 0 2 2 -2 -2 0 0 0 0 0 0 2 -2 2 0 0 -2 2 0 0 -2 0 0 orthogonal lifted from D4 ρ15 2 2 2 0 0 2 -2 -2 0 0 0 0 -2 2 0 2 2 0 0 0 -2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 -2 0 -1 2 2 -2 -2 0 0 0 0 0 -1 -1 1 2 0 -1 -1 1 1 1 -1 -1 orthogonal lifted from D6 ρ17 2 2 2 2 0 -1 2 2 2 -2 0 0 0 0 0 -1 -1 -1 -2 0 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ18 2 2 2 0 0 2 -2 -2 0 0 0 0 2 -2 0 2 2 0 0 0 -2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 0 -1 2 -2 2 0 0 0 0 0 0 -1 1 1 0 0 1 -1 -√-3 √-3 -1 -√-3 √-3 complex lifted from C3⋊D4 ρ20 2 2 -2 2 0 -1 2 -2 -2 0 0 0 0 0 0 -1 1 -1 0 0 1 -1 -√-3 √-3 1 √-3 -√-3 complex lifted from C3⋊D4 ρ21 2 2 -2 2 0 -1 2 -2 -2 0 0 0 0 0 0 -1 1 -1 0 0 1 -1 √-3 -√-3 1 -√-3 √-3 complex lifted from C3⋊D4 ρ22 2 2 -2 -2 0 -1 2 -2 2 0 0 0 0 0 0 -1 1 1 0 0 1 -1 √-3 -√-3 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ23 4 4 -4 0 0 -2 -4 4 0 0 0 0 0 0 0 -2 2 0 0 0 -2 2 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 0 2 2 0 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 0 4 0 0 0 0 -2 0 0 0 2 -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 0 -2 -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 D12.40D4
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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
(1 46 10 37 7 40 4 43)(2 41 11 44 8 47 5 38)(3 48 12 39 9 42 6 45)(13 34 22 25 19 28 16 31)(14 29 23 32 20 35 17 26)(15 36 24 27 21 30 18 33)
(1 25 7 31)(2 26 8 32)(3 27 9 33)(4 28 10 34)(5 29 11 35)(6 30 12 36)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)```

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

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

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

Matrix representation of D12.40D4 in GL8(𝔽73)

 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 34 39 72 70 0 0 0 0 26 0 25 1
,
 24 49 69 4 0 0 0 0 25 49 8 4 0 0 0 0 4 69 49 24 0 0 0 0 65 69 48 24 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 42 31 46 65 0 0 0 0 46 0 0 0 0 0 0 0 44 47 28 42
,
 49 25 4 8 0 0 0 0 48 24 65 69 0 0 0 0 69 65 24 48 0 0 0 0 8 4 25 49 0 0 0 0 0 0 0 0 42 31 46 65 0 0 0 0 0 0 46 0 0 0 0 0 27 0 0 0 0 0 0 0 29 26 0 31
,
 69 65 24 48 0 0 0 0 8 4 25 49 0 0 0 0 49 25 4 8 0 0 0 0 48 24 65 69 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 39 34 1 3 0 0 0 0 72 0 0 0 0 0 0 0 45 28 0 39

`G:=sub<GL(8,GF(73))| [1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,34,26,0,0,0,0,72,0,39,0,0,0,0,0,0,0,72,25,0,0,0,0,0,0,70,1],[24,25,4,65,0,0,0,0,49,49,69,69,0,0,0,0,69,8,49,48,0,0,0,0,4,4,24,24,0,0,0,0,0,0,0,0,0,42,46,44,0,0,0,0,0,31,0,47,0,0,0,0,27,46,0,28,0,0,0,0,0,65,0,42],[49,48,69,8,0,0,0,0,25,24,65,4,0,0,0,0,4,65,24,25,0,0,0,0,8,69,48,49,0,0,0,0,0,0,0,0,42,0,27,29,0,0,0,0,31,0,0,26,0,0,0,0,46,46,0,0,0,0,0,0,65,0,0,31],[69,8,49,48,0,0,0,0,65,4,25,24,0,0,0,0,24,25,4,65,0,0,0,0,48,49,8,69,0,0,0,0,0,0,0,0,0,39,72,45,0,0,0,0,0,34,0,28,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,39] >;`

D12.40D4 in GAP, Magma, Sage, TeX

`D_{12}._{40}D_4`
`% in TeX`

`G:=Group("D12.40D4");`
`// GroupNames label`

`G:=SmallGroup(192,764);`
`// by ID`

`G=gap.SmallGroup(192,764);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,184,1123,297,136,851,438,102,6278]);`
`// Polycyclic`

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

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