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## G = D12.15D4order 192 = 26·3

### 15th non-split extension by D12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

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

Generators and relations for D12.15D4
G = < a,b,c,d | a12=b2=d2=1, c4=a6, bab=cac-1=a-1, dad=a5, cbc-1=a7b, dbd=a10b, dcd=c3 >

Subgroups: 400 in 142 conjugacy classes, 39 normal (19 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, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C4.Dic3, Q82S3, C3⋊Q16, C4×C12, C3×C4⋊C4, C4○D12, C4○D12, S3×Q8, Q83S3, C6×Q8, D4.10D4, C424S3, C12.10D4, Q8.11D6, C3×C4⋊Q8, Q8.15D6, D12.15D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, D4.10D4, C232D6, D12.15D4

Character table of D12.15D4

 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 12F 12G 12H 12I 12J size 1 1 2 12 12 2 2 2 4 4 4 4 8 12 12 2 2 2 24 24 4 4 4 4 4 4 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 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 0 -2 2 2 -2 0 0 0 0 0 2 0 -2 2 -2 0 0 0 0 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 2 2 2 -2 0 0 0 0 0 -2 0 -2 2 -2 0 0 0 0 0 -2 2 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 0 2 -2 2 0 0 0 0 0 0 -2 -2 2 -2 0 0 0 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 0 0 2 -2 -2 2 0 0 -2 0 0 0 2 2 2 0 0 0 0 0 -2 -2 0 0 2 0 -2 orthogonal lifted from D4 ρ13 2 2 2 0 0 2 -2 -2 -2 0 0 2 0 0 0 2 2 2 0 0 0 0 0 -2 -2 0 0 -2 0 2 orthogonal lifted from D4 ρ14 2 2 -2 -2 0 2 -2 2 0 0 0 0 0 0 2 -2 2 -2 0 0 0 0 0 2 -2 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 0 0 -1 2 2 -2 2 2 -2 -2 0 0 -1 -1 -1 0 0 -1 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ16 2 2 2 0 0 -1 2 2 2 2 2 2 2 0 0 -1 -1 -1 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ17 2 2 2 0 0 -1 2 2 -2 -2 -2 -2 2 0 0 -1 -1 -1 0 0 1 1 1 -1 -1 1 -1 1 -1 1 orthogonal lifted from D6 ρ18 2 2 2 0 0 -1 2 2 2 -2 -2 2 -2 0 0 -1 -1 -1 0 0 1 1 1 -1 -1 1 1 -1 1 -1 orthogonal lifted from D6 ρ19 2 2 2 0 0 -1 -2 -2 2 0 0 -2 0 0 0 -1 -1 -1 0 0 -√-3 √-3 -√-3 1 1 √-3 √-3 -1 -√-3 1 complex lifted from C3⋊D4 ρ20 2 2 2 0 0 -1 -2 -2 2 0 0 -2 0 0 0 -1 -1 -1 0 0 √-3 -√-3 √-3 1 1 -√-3 -√-3 -1 √-3 1 complex lifted from C3⋊D4 ρ21 2 2 2 0 0 -1 -2 -2 -2 0 0 2 0 0 0 -1 -1 -1 0 0 √-3 -√-3 √-3 1 1 -√-3 √-3 1 -√-3 -1 complex lifted from C3⋊D4 ρ22 2 2 2 0 0 -1 -2 -2 -2 0 0 2 0 0 0 -1 -1 -1 0 0 -√-3 √-3 -√-3 1 1 √-3 -√-3 1 √-3 -1 complex lifted from C3⋊D4 ρ23 4 4 -4 0 0 -2 -4 4 0 0 0 0 0 0 0 2 -2 2 0 0 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 2 0 0 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 -2 2 0 0 0 0 0 -4 0 0 0 -2 2 2 0 0 -2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ26 4 -4 0 0 0 4 0 0 0 2 -2 0 0 0 0 0 -4 0 0 0 2 -2 -2 0 0 2 0 0 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ27 4 -4 0 0 0 -2 0 0 0 2 -2 0 0 0 0 -2√-3 2 2√-3 0 0 -1-√-3 1-√-3 1+√-3 0 0 -1+√-3 0 0 0 0 complex faithful ρ28 4 -4 0 0 0 -2 0 0 0 -2 2 0 0 0 0 -2√-3 2 2√-3 0 0 1+√-3 -1+√-3 -1-√-3 0 0 1-√-3 0 0 0 0 complex faithful ρ29 4 -4 0 0 0 -2 0 0 0 -2 2 0 0 0 0 2√-3 2 -2√-3 0 0 1-√-3 -1-√-3 -1+√-3 0 0 1+√-3 0 0 0 0 complex faithful ρ30 4 -4 0 0 0 -2 0 0 0 2 -2 0 0 0 0 2√-3 2 -2√-3 0 0 -1+√-3 1+√-3 1-√-3 0 0 -1-√-3 0 0 0 0 complex faithful

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

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

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

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

Matrix representation of D12.15D4 in GL4(𝔽7) generated by

 3 4 3 6 3 5 3 1 5 3 5 6 6 5 1 1
,
 2 6 0 2 2 3 6 0 3 6 4 4 3 6 3 5
,
 2 4 4 5 6 3 3 6 3 0 1 4 1 2 5 1
,
 3 6 6 6 2 3 0 4 2 6 0 6 4 2 4 1
`G:=sub<GL(4,GF(7))| [3,3,5,6,4,5,3,5,3,3,5,1,6,1,6,1],[2,2,3,3,6,3,6,6,0,6,4,3,2,0,4,5],[2,6,3,1,4,3,0,2,4,3,1,5,5,6,4,1],[3,2,2,4,6,3,6,2,6,0,0,4,6,4,6,1] >;`

D12.15D4 in GAP, Magma, Sage, TeX

`D_{12}._{15}D_4`
`% in TeX`

`G:=Group("D12.15D4");`
`// GroupNames label`

`G:=SmallGroup(192,654);`
`// by ID`

`G=gap.SmallGroup(192,654);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,184,1123,570,297,136,1684,6278]);`
`// 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=c^3>;`
`// generators/relations`

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