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## G = D12.14D4order 192 = 26·3

### 14th non-split extension by D12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.14D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — Q8.15D6 — D12.14D4
 Lower central C3 — C6 — C2×C12 — D12.14D4
 Upper central C1 — C2 — C2×C4 — C4.4D4

Generators and relations for D12.14D4
G = < a,b,c,d | a12=b2=d2=1, c4=a6, bab=cac-1=a-1, dad=a5, cbc-1=a7b, dbd=a10b, dcd=a6c3 >

Subgroups: 432 in 146 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×5], S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×7], D4 [×8], Q8 [×6], C23, Dic3 [×2], C12 [×2], C12 [×3], D6 [×2], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C4○D4 [×6], C3⋊C8 [×2], Dic6 [×2], Dic6 [×2], C4×S3 [×6], D12 [×2], D12 [×2], C3⋊D4 [×2], C2×C12, C2×C12 [×2], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×C6, C4.10D4, C4≀C2 [×2], C4.4D4, C8⋊C22 [×2], 2- 1+4, C4.Dic3 [×2], D4⋊S3 [×2], D4.S3 [×2], C4×C12, C3×C22⋊C4 [×2], C4○D12 [×2], C4○D12 [×2], S3×Q8 [×2], Q83S3 [×2], C6×D4, C6×Q8, D4.8D4, C424S3 [×2], C12.10D4, D126C22 [×2], C3×C4.4D4, Q8.15D6, D12.14D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, S3×D4 [×2], C2×C3⋊D4, D4.8D4, C232D6, D12.14D4

Character table of D12.14D4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 2 8 12 12 2 2 2 4 4 4 4 12 12 2 2 2 8 8 24 24 4 4 4 4 4 4 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 -2 0 0 -1 -1 -1 1 1 0 0 -1 -1 -1 -1 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 2 -2 0 -2 0 2 -2 2 0 0 0 0 2 0 -2 2 -2 0 0 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 2 0 2 -2 2 0 0 0 0 -2 0 -2 2 -2 0 0 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 -2 0 0 -2 2 2 -2 0 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 -1 2 2 -2 -2 -2 -2 0 0 -1 -1 -1 -1 -1 0 0 1 1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ14 2 2 -2 0 0 2 2 2 -2 0 0 0 0 0 -2 -2 2 -2 0 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 0 0 0 2 -2 -2 0 2 0 -2 0 0 2 2 2 0 0 0 0 0 0 -2 -2 0 0 2 -2 orthogonal lifted from D4 ρ16 2 2 2 0 0 0 2 -2 -2 0 -2 0 2 0 0 2 2 2 0 0 0 0 0 0 -2 -2 0 0 -2 2 orthogonal lifted from D4 ρ17 2 2 2 2 0 0 -1 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ18 2 2 2 -2 0 0 -1 2 2 -2 2 -2 2 0 0 -1 -1 -1 1 1 0 0 1 1 -1 -1 1 1 -1 -1 orthogonal lifted from D6 ρ19 2 2 2 0 0 0 -1 -2 -2 0 2 0 -2 0 0 -1 -1 -1 √-3 -√-3 0 0 -√-3 √-3 1 1 -√-3 √-3 -1 1 complex lifted from C3⋊D4 ρ20 2 2 2 0 0 0 -1 -2 -2 0 -2 0 2 0 0 -1 -1 -1 √-3 -√-3 0 0 √-3 -√-3 1 1 √-3 -√-3 1 -1 complex lifted from C3⋊D4 ρ21 2 2 2 0 0 0 -1 -2 -2 0 2 0 -2 0 0 -1 -1 -1 -√-3 √-3 0 0 √-3 -√-3 1 1 √-3 -√-3 -1 1 complex lifted from C3⋊D4 ρ22 2 2 2 0 0 0 -1 -2 -2 0 -2 0 2 0 0 -1 -1 -1 -√-3 √-3 0 0 -√-3 √-3 1 1 -√-3 √-3 1 -1 complex lifted from C3⋊D4 ρ23 4 4 -4 0 0 0 -2 4 -4 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 -2 2 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 2 0 0 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 0 0 4 0 0 -2i 0 2i 0 0 0 0 -4 0 0 0 0 0 -2i -2i 0 0 2i 2i 0 0 complex lifted from D4.8D4 ρ26 4 -4 0 0 0 0 4 0 0 2i 0 -2i 0 0 0 0 -4 0 0 0 0 0 2i 2i 0 0 -2i -2i 0 0 complex lifted from D4.8D4 ρ27 4 -4 0 0 0 0 -2 0 0 2i 0 -2i 0 0 0 2√-3 2 -2√-3 0 0 0 0 2ζ4ζ32 2ζ4ζ3 0 0 2ζ43ζ32 2ζ43ζ3 0 0 complex faithful ρ28 4 -4 0 0 0 0 -2 0 0 -2i 0 2i 0 0 0 -2√-3 2 2√-3 0 0 0 0 2ζ43ζ3 2ζ43ζ32 0 0 2ζ4ζ3 2ζ4ζ32 0 0 complex faithful ρ29 4 -4 0 0 0 0 -2 0 0 -2i 0 2i 0 0 0 2√-3 2 -2√-3 0 0 0 0 2ζ43ζ32 2ζ43ζ3 0 0 2ζ4ζ32 2ζ4ζ3 0 0 complex faithful ρ30 4 -4 0 0 0 0 -2 0 0 2i 0 -2i 0 0 0 -2√-3 2 2√-3 0 0 0 0 2ζ4ζ3 2ζ4ζ32 0 0 2ζ43ζ3 2ζ43ζ32 0 0 complex faithful

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

Matrix representation of D12.14D4 in GL4(𝔽73) generated by

 0 9 0 0 64 0 0 0 0 0 0 65 0 0 8 0
,
 0 0 70 0 0 0 0 3 24 0 0 0 0 49 0 0
,
 0 0 0 72 0 0 72 0 1 0 0 0 0 72 0 0
,
 0 0 0 1 0 0 72 0 0 72 0 0 1 0 0 0
`G:=sub<GL(4,GF(73))| [0,64,0,0,9,0,0,0,0,0,0,8,0,0,65,0],[0,0,24,0,0,0,0,49,70,0,0,0,0,3,0,0],[0,0,1,0,0,0,0,72,0,72,0,0,72,0,0,0],[0,0,0,1,0,0,72,0,0,72,0,0,1,0,0,0] >;`

D12.14D4 in GAP, Magma, Sage, TeX

`D_{12}._{14}D_4`
`% in TeX`

`G:=Group("D12.14D4");`
`// GroupNames label`

`G:=SmallGroup(192,621);`
`// by ID`

`G=gap.SmallGroup(192,621);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,1123,570,297,136,1684,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=d^2=1,c^4=a^6,b*a*b=c*a*c^-1=a^-1,d*a*d=a^5,c*b*c^-1=a^7*b,d*b*d=a^10*b,d*c*d=a^6*c^3>;`
`// generators/relations`

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