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## G = D8.D6order 192 = 26·3

### 1st non-split extension by D8 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — D8.D6
 Chief series C1 — C3 — C6 — C12 — C24 — D24 — C4○D24 — D8.D6
 Lower central C3 — C6 — C12 — C24 — D8.D6
 Upper central C1 — C2 — C2×C4 — C2×C8 — C2×D8

Generators and relations for D8.D6
G = < a,b,c,d | a8=b2=1, c6=d2=a4, bab=dad-1=a-1, ac=ca, cbc-1=a4b, dbd-1=ab, dcd-1=c5 >

Subgroups: 312 in 90 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C2×C6, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22×C6, M5(2), D16, SD32, C2×D8, C4○D8, C3⋊C16, C24⋊C2, D24, Dic12, C2×C24, C3×D8, C3×D8, C4○D12, C6×D4, C16⋊C22, C12.C8, C3⋊D16, D8.S3, C4○D24, C6×D8, D8.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C2×D8, D4⋊S3, C2×C3⋊D4, C16⋊C22, C2×D4⋊S3, D8.D6

Character table of D8.D6

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 12A 12B 16A 16B 16C 16D 24A 24B 24C 24D size 1 1 2 8 8 24 2 2 2 24 2 2 2 8 8 8 8 2 2 4 4 4 12 12 12 12 4 4 4 4 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 2 0 -1 2 2 0 -1 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 -2 2 0 -1 -2 2 0 -1 1 1 1 -1 -1 1 2 2 -2 -1 1 0 0 0 0 1 -1 1 -1 orthogonal lifted from D6 ρ11 2 2 -2 2 -2 0 -1 -2 2 0 -1 1 1 -1 1 1 -1 2 2 -2 -1 1 0 0 0 0 1 -1 1 -1 orthogonal lifted from D6 ρ12 2 2 2 -2 -2 0 -1 2 2 0 -1 -1 -1 1 1 1 1 2 2 2 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from D6 ρ13 2 2 2 0 0 0 2 2 2 0 2 2 2 0 0 0 0 -2 -2 -2 2 2 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ14 2 2 -2 0 0 0 2 -2 2 0 2 -2 -2 0 0 0 0 -2 -2 2 2 -2 0 0 0 0 2 -2 2 -2 orthogonal lifted from D4 ρ15 2 2 -2 0 0 0 2 2 -2 0 2 -2 -2 0 0 0 0 0 0 0 -2 2 -√2 √2 -√2 √2 0 0 0 0 orthogonal lifted from D8 ρ16 2 2 -2 0 0 0 2 2 -2 0 2 -2 -2 0 0 0 0 0 0 0 -2 2 √2 -√2 √2 -√2 0 0 0 0 orthogonal lifted from D8 ρ17 2 2 2 0 0 0 2 -2 -2 0 2 2 2 0 0 0 0 0 0 0 -2 -2 √2 √2 -√2 -√2 0 0 0 0 orthogonal lifted from D8 ρ18 2 2 2 0 0 0 2 -2 -2 0 2 2 2 0 0 0 0 0 0 0 -2 -2 -√2 -√2 √2 √2 0 0 0 0 orthogonal lifted from D8 ρ19 2 2 -2 0 0 0 -1 -2 2 0 -1 1 1 -√-3 -√-3 √-3 √-3 -2 -2 2 -1 1 0 0 0 0 -1 1 -1 1 complex lifted from C3⋊D4 ρ20 2 2 2 0 0 0 -1 2 2 0 -1 -1 -1 -√-3 √-3 -√-3 √-3 -2 -2 -2 -1 -1 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ21 2 2 2 0 0 0 -1 2 2 0 -1 -1 -1 √-3 -√-3 √-3 -√-3 -2 -2 -2 -1 -1 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ22 2 2 -2 0 0 0 -1 -2 2 0 -1 1 1 √-3 √-3 -√-3 -√-3 -2 -2 2 -1 1 0 0 0 0 -1 1 -1 1 complex lifted from C3⋊D4 ρ23 4 4 -4 0 0 0 -2 4 -4 0 -2 2 2 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ24 4 4 4 0 0 0 -2 -4 -4 0 -2 -2 -2 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ25 4 -4 0 0 0 0 4 0 0 0 -4 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 0 -2√2 0 2√2 orthogonal lifted from C16⋊C22 ρ26 4 -4 0 0 0 0 4 0 0 0 -4 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 0 2√2 0 -2√2 orthogonal lifted from C16⋊C22 ρ27 4 -4 0 0 0 0 -2 0 0 0 2 2√-3 -2√-3 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 √-6 -√2 -√-6 √2 complex faithful ρ28 4 -4 0 0 0 0 -2 0 0 0 2 -2√-3 2√-3 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 √-6 √2 -√-6 -√2 complex faithful ρ29 4 -4 0 0 0 0 -2 0 0 0 2 -2√-3 2√-3 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 -√-6 -√2 √-6 √2 complex faithful ρ30 4 -4 0 0 0 0 -2 0 0 0 2 2√-3 -2√-3 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 -√-6 √2 √-6 -√2 complex faithful

Smallest permutation representation of D8.D6
On 48 points
Generators in S48
```(1 22 10 19 7 16 4 13)(2 23 11 20 8 17 5 14)(3 24 12 21 9 18 6 15)(25 37 28 40 31 43 34 46)(26 38 29 41 32 44 35 47)(27 39 30 42 33 45 36 48)
(1 13)(2 20)(3 15)(4 22)(5 17)(6 24)(7 19)(8 14)(9 21)(10 16)(11 23)(12 18)(25 34)(26 29)(27 36)(28 31)(30 33)(32 35)(37 43)(39 45)(41 47)
(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 34 7 28)(2 27 8 33)(3 32 9 26)(4 25 10 31)(5 30 11 36)(6 35 12 29)(13 46 19 40)(14 39 20 45)(15 44 21 38)(16 37 22 43)(17 42 23 48)(18 47 24 41)```

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

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

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

Matrix representation of D8.D6 in GL4(𝔽7) generated by

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

D8.D6 in GAP, Magma, Sage, TeX

`D_8.D_6`
`% in TeX`

`G:=Group("D8.D6");`
`// GroupNames label`

`G:=SmallGroup(192,706);`
`// by ID`

`G=gap.SmallGroup(192,706);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,387,675,185,192,1684,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=1,c^6=d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^5>;`
`// generators/relations`

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