Copied to
clipboard

## G = D8⋊5D6order 192 = 26·3

### 5th semidirect product of D8 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D8⋊5D6
 Chief series C1 — C3 — C6 — C12 — C4×S3 — C4○D12 — D4⋊6D6 — D8⋊5D6
 Lower central C3 — C6 — C12 — D8⋊5D6
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D85D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=a-1, cac-1=a5, dad=a3, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 832 in 268 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C2×D4, C2×D4, C4○D4, C4○D4, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C8○D4, C2×D8, C4○D8, C8⋊C22, C8⋊C22, 2+ 1+4, S3×C8, C8⋊S3, C24⋊C2, D24, C2×C3⋊C8, D4⋊S3, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C3×M4(2), C3×D8, C3×SD16, C2×D12, C2×D12, C4○D12, C4○D12, S3×D4, S3×D4, D42S3, D42S3, Q83S3, C2×C3⋊D4, C6×D4, C3×C4○D4, D4○D8, D12.C4, C8⋊D6, S3×D8, D8⋊S3, Q83D6, Q8.7D6, C2×D4⋊S3, Q8.13D6, C3×C8⋊C22, D46D6, D4○D12, D85D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, D4○D8, C2×S3×D4, D85D6

Smallest permutation representation of D85D6
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)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(18 24)(19 23)(20 22)(25 31)(26 30)(27 29)(33 35)(36 40)(37 39)(41 43)(44 48)(45 47)
(1 46 32 21 38 14)(2 43 25 18 39 11)(3 48 26 23 40 16)(4 45 27 20 33 13)(5 42 28 17 34 10)(6 47 29 22 35 15)(7 44 30 19 36 12)(8 41 31 24 37 9)
(1 11)(2 14)(3 9)(4 12)(5 15)(6 10)(7 13)(8 16)(17 29)(18 32)(19 27)(20 30)(21 25)(22 28)(23 31)(24 26)(33 44)(34 47)(35 42)(36 45)(37 48)(38 43)(39 46)(40 41)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E 12A 12B 12C 24A 24B order 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 8 12 12 12 24 24 size 1 1 2 4 4 4 6 6 12 12 12 2 2 2 4 6 6 12 2 4 8 8 8 4 4 6 6 12 4 4 8 8 8

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D6 D6 S3×D4 S3×D4 D4○D8 D8⋊5D6 kernel D8⋊5D6 D12.C4 C8⋊D6 S3×D8 D8⋊S3 Q8⋊3D6 Q8.7D6 C2×D4⋊S3 Q8.13D6 C3×C8⋊C22 D4⋊6D6 D4○D12 C8⋊C22 Dic6 D12 C3⋊D4 M4(2) D8 SD16 C2×D4 C4○D4 C4 C22 C3 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 1 2 2 1 1 1 1 2 1

Matrix representation of D85D6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 16 57 0 0 0 0 16 16 0 0 0 0 0 0 57 16 0 0 0 0 57 57
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 1 72 0 0 0 0 0 72 0 0 0 0 0 0 0 0 16 16 0 0 0 0 16 57 0 0 16 16 0 0 0 0 16 57 0 0

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,16,16,0,0,0,0,57,16,0,0,0,0,0,0,57,57,0,0,0,0,16,57],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,16,16,0,0,0,0,16,57,0,0,16,16,0,0,0,0,16,57,0,0] >;`

D85D6 in GAP, Magma, Sage, TeX

`D_8\rtimes_5D_6`
`% in TeX`

`G:=Group("D8:5D6");`
`// GroupNames label`

`G:=SmallGroup(192,1333);`
`// by ID`

`G=gap.SmallGroup(192,1333);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,570,185,438,235,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=a^5,d*a*d=a^3,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽