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## G = D12.39D4order 192 = 26·3

### 9th non-split extension by D12 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.39D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — D4○D12 — D12.39D4
 Lower central C3 — C6 — C2×C12 — D12.39D4
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for D12.39D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=ab, dbd=a10b, dcd=a6c-1 >

Subgroups: 528 in 152 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, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, M4(2), M4(2), SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×S3, C4.D4, C4≀C2, C4.4D4, C8.C22, C8.C22, 2+ 1+4, C4.Dic3, C4×Dic3, D6⋊C4, Q82S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C2×D12, C2×D12, C4○D12, C4○D12, S3×D4, Q83S3, C6×Q8, C3×C4○D4, D4.9D4, C12.46D4, D12⋊C4, Q83Dic3, Q8.11D6, C12.23D4, C3×C8.C22, D4○D12, D12.39D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, D4.9D4, C232D6, D12.39D4

Character table of D12.39D4

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 24A 24B size 1 1 2 4 12 12 12 2 2 2 4 8 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 0 -2 2 2 -2 -2 0 0 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 0 0 2 2 -2 2 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 0 0 -1 2 2 2 -2 0 0 0 -1 -1 -1 -2 0 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ12 2 2 -2 0 2 0 0 2 -2 2 0 0 -2 0 0 2 -2 0 0 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 2 0 0 0 2 2 -2 -2 0 0 0 0 2 -2 2 0 0 2 -2 0 0 -2 0 0 orthogonal lifted from D4 ρ14 2 2 2 0 0 2 -2 2 -2 -2 0 0 0 0 0 2 2 0 0 0 -2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 -2 0 0 0 -1 2 2 -2 -2 0 0 0 -1 -1 1 2 0 -1 -1 1 1 1 -1 -1 orthogonal lifted from D6 ρ16 2 2 2 2 0 0 0 -1 2 2 2 2 0 0 0 -1 -1 -1 2 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ17 2 2 2 -2 0 0 0 -1 2 2 -2 2 0 0 0 -1 -1 1 -2 0 -1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ18 2 2 -2 0 -2 0 0 2 -2 2 0 0 2 0 0 2 -2 0 0 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 0 0 0 -1 2 -2 2 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 0 0 -1 2 -2 -2 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 0 0 -1 2 -2 -2 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 0 0 -1 2 -2 2 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 0 0 -2 -4 4 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 0 0 -2 -4 -4 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 0 0 4 0 0 0 0 0 -2i 2i -4 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.9D4 ρ26 4 -4 0 0 0 0 0 4 0 0 0 0 0 2i -2i -4 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.9D4 ρ27 8 -8 0 0 0 0 0 -4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2

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

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

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

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

Matrix representation of D12.39D4 in GL8(𝔽73)

 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 1 0 0 72 0 0 0 0 72 0 72 0 0 0 0 0 72 1 0 1 0 0 0 0 2 0 0 72
,
 1 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 2 1 72 0 0 0 0 2 1 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 72 72 0 0 0 0 1 72 0 72 0 0 0 0 71 0 0 1
,
 1 0 48 48 0 0 0 0 72 72 48 50 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 46 27 0 27 0 0 0 0 0 46 0 0 0 0 0 0 0 46 46 46 0 0 0 0 46 27 27 27
,
 72 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 1 72 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 71 72

`G:=sub<GL(8,GF(73))| [1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,72,72,2,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,1,72],[1,0,1,2,0,0,0,0,1,72,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,72,1,1,71,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,72,1],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,48,48,0,72,0,0,0,0,48,50,72,0,0,0,0,0,0,0,0,0,46,0,0,46,0,0,0,0,27,46,46,27,0,0,0,0,0,0,46,27,0,0,0,0,27,0,46,27],[72,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,1,1,71,0,0,0,0,0,0,0,72] >;`

D12.39D4 in GAP, Magma, Sage, TeX

`D_{12}._{39}D_4`
`% in TeX`

`G:=Group("D12.39D4");`
`// GroupNames label`

`G:=SmallGroup(192,761);`
`// by ID`

`G=gap.SmallGroup(192,761);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,184,570,1684,851,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a*b,d*b*d=a^10*b,d*c*d=a^6*c^-1>;`
`// generators/relations`

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