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## G = D4.10D12order 192 = 26·3

### 5th non-split extension by D4 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4.10D12
G = < a,b,c,d | a12=b2=c4=1, d2=a6, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a9c-1 >

Subgroups: 480 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×5], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×9], D4, D4 [×7], Q8, Q8 [×5], C23, Dic3 [×3], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6, C42, C22⋊C4 [×2], M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4, C2×Q8 [×3], C4○D4, C4○D4 [×5], C3⋊C8, C24, Dic6, Dic6 [×4], C4×S3 [×3], D12, D12 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C22×S3, C4.10D4, C4≀C2, C4≀C2, C4.4D4, C8⋊C22 [×2], 2- 1+4, C24⋊C2, D24, C4.Dic3, D6⋊C4 [×2], D4⋊S3, Q82S3, C4×C12, C3×M4(2), C2×Dic6, C2×Dic6, C2×D12, C4○D12, C4○D12, D42S3 [×3], S3×Q8, C3×C4○D4, D4.8D4, C424S3, C12.47D4, C3×C4≀C2, C427S3, C8⋊D6, D4⋊D6, Q8○D12, D4.10D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×2], C22×S3, C22≀C2, C2×D12, S3×D4 [×2], D4.8D4, D6⋊D4, D4.10D12

Character table of D4.10D12

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 24A 24B size 1 1 2 4 12 24 2 2 2 4 4 4 12 12 12 2 4 8 8 24 2 2 4 4 4 4 4 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 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 -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 -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 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 -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 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 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 -1 1 1 linear of order 2 ρ9 2 2 2 -2 0 0 -1 2 2 -2 -2 -2 0 0 0 -1 -1 1 2 0 -1 -1 1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ10 2 2 2 -2 0 0 -1 2 2 -2 2 2 0 0 0 -1 -1 1 -2 0 -1 -1 -1 -1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ11 2 2 2 0 0 0 2 -2 -2 0 0 0 2 0 -2 2 2 0 0 0 -2 -2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 2 0 0 2 -2 2 -2 0 0 0 0 0 2 -2 2 0 0 -2 -2 0 2 0 0 0 -2 0 0 orthogonal lifted from D4 ρ13 2 2 -2 -2 0 0 2 -2 2 2 0 0 0 0 0 2 -2 -2 0 0 -2 -2 0 2 0 0 0 2 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 2 0 2 2 -2 0 0 0 0 -2 0 2 -2 0 0 0 2 2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 0 0 0 2 -2 -2 0 0 0 -2 0 2 2 2 0 0 0 -2 -2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 0 0 -1 2 2 2 -2 -2 0 0 0 -1 -1 -1 -2 0 -1 -1 1 -1 1 1 1 -1 1 1 orthogonal lifted from D6 ρ17 2 2 2 2 0 0 -1 2 2 2 2 2 0 0 0 -1 -1 -1 2 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ18 2 2 -2 0 -2 0 2 2 -2 0 0 0 0 2 0 2 -2 0 0 0 2 2 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 2 0 0 -1 -2 2 -2 0 0 0 0 0 -1 1 -1 0 0 1 1 -√3 -1 √3 -√3 √3 1 -√3 √3 orthogonal lifted from D12 ρ20 2 2 -2 -2 0 0 -1 -2 2 2 0 0 0 0 0 -1 1 1 0 0 1 1 -√3 -1 √3 -√3 √3 -1 √3 -√3 orthogonal lifted from D12 ρ21 2 2 -2 2 0 0 -1 -2 2 -2 0 0 0 0 0 -1 1 -1 0 0 1 1 √3 -1 -√3 √3 -√3 1 √3 -√3 orthogonal lifted from D12 ρ22 2 2 -2 -2 0 0 -1 -2 2 2 0 0 0 0 0 -1 1 1 0 0 1 1 √3 -1 -√3 √3 -√3 -1 -√3 √3 orthogonal lifted from D12 ρ23 4 4 4 0 0 0 -2 -4 -4 0 0 0 0 0 0 -2 -2 0 0 0 2 2 0 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 -4 0 0 0 -2 4 -4 0 0 0 0 0 0 -2 2 0 0 0 -2 -2 0 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 0 0 4 0 0 0 -2i 2i 0 0 0 -4 0 0 0 0 0 0 -2i 0 2i 2i -2i 0 0 0 complex lifted from D4.8D4 ρ26 4 -4 0 0 0 0 4 0 0 0 2i -2i 0 0 0 -4 0 0 0 0 0 0 2i 0 -2i -2i 2i 0 0 0 complex lifted from D4.8D4 ρ27 4 -4 0 0 0 0 -2 0 0 0 -2i 2i 0 0 0 2 0 0 0 0 -2√3 2√3 ζ4+2ζ32+1 0 ζ43+2ζ32+1 ζ43+2ζ3+1 ζ4+2ζ3+1 0 0 0 complex faithful ρ28 4 -4 0 0 0 0 -2 0 0 0 2i -2i 0 0 0 2 0 0 0 0 -2√3 2√3 ζ43+2ζ3+1 0 ζ4+2ζ3+1 ζ4+2ζ32+1 ζ43+2ζ32+1 0 0 0 complex faithful ρ29 4 -4 0 0 0 0 -2 0 0 0 -2i 2i 0 0 0 2 0 0 0 0 2√3 -2√3 ζ4+2ζ3+1 0 ζ43+2ζ3+1 ζ43+2ζ32+1 ζ4+2ζ32+1 0 0 0 complex faithful ρ30 4 -4 0 0 0 0 -2 0 0 0 2i -2i 0 0 0 2 0 0 0 0 2√3 -2√3 ζ43+2ζ32+1 0 ζ4+2ζ32+1 ζ4+2ζ3+1 ζ43+2ζ3+1 0 0 0 complex faithful

Smallest permutation representation of D4.10D12
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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 34 19 28)(14 35 20 29)(15 36 21 30)(16 25 22 31)(17 26 23 32)(18 27 24 33)
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,34,19,28)(14,35,20,29)(15,36,21,30)(16,25,22,31)(17,26,23,32)(18,27,24,33), (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,34,19,28)(14,35,20,29)(15,36,21,30)(16,25,22,31)(17,26,23,32)(18,27,24,33), (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(25,42),(26,41),(27,40),(28,39),(29,38),(30,37),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,34,19,28),(14,35,20,29),(15,36,21,30),(16,25,22,31),(17,26,23,32),(18,27,24,33)], [(1,31,7,25),(2,32,8,26),(3,33,9,27),(4,34,10,28),(5,35,11,29),(6,36,12,30),(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 D4.10D12 in GL6(𝔽73)

 63 52 0 0 0 0 47 11 0 0 0 0 0 0 72 72 0 1 0 0 2 1 72 72 0 0 0 0 0 1 0 0 0 0 72 0
,
 63 1 0 0 0 0 47 10 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 71 72 1 1
,
 72 71 0 0 0 0 1 1 0 0 0 0 0 0 27 27 0 0 0 0 19 46 27 27 0 0 0 0 27 0 0 0 0 0 0 27
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 46 27 0 0 0 0 0 46 0 0 0 0 46 0 0 0 0 19 46 27 27

`G:=sub<GL(6,GF(73))| [63,47,0,0,0,0,52,11,0,0,0,0,0,0,72,2,0,0,0,0,72,1,0,0,0,0,0,72,0,72,0,0,1,72,1,0],[63,47,0,0,0,0,1,10,0,0,0,0,0,0,72,0,0,71,0,0,0,0,1,72,0,0,0,1,0,1,0,0,0,0,0,1],[72,1,0,0,0,0,71,1,0,0,0,0,0,0,27,19,0,0,0,0,27,46,0,0,0,0,0,27,27,0,0,0,0,27,0,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,19,0,0,46,0,46,46,0,0,27,46,0,27,0,0,0,0,0,27] >;`

D4.10D12 in GAP, Magma, Sage, TeX

`D_4._{10}D_{12}`
`% in TeX`

`G:=Group("D4.10D12");`
`// GroupNames label`

`G:=SmallGroup(192,386);`
`// by ID`

`G=gap.SmallGroup(192,386);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,226,1123,136,851,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^9*c^-1>;`
`// generators/relations`

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