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## G = D8⋊4D6order 192 = 26·3

### 4th semidirect product of D8 and D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D8⋊4D6
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — S3×C4○D4 — D8⋊4D6
 Lower central C3 — C6 — C12 — D8⋊4D6
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D84D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=a-1, cac-1=a5, ad=da, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 688 in 262 conjugacy classes, 99 normal (51 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, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C24, Dic6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C2×M4(2), C4○D8, C8⋊C22, C8⋊C22, C8.C22, C2×C4○D4, S3×C8, C8⋊S3, C24⋊C2, Dic12, C4.Dic3, D4⋊S3, D4.S3, C3⋊Q16, C3×M4(2), C3×D8, C3×SD16, C2×Dic6, S3×C2×C4, S3×C2×C4, C4○D12, C4○D12, S3×D4, S3×D4, D42S3, D42S3, D42S3, S3×Q8, Q83S3, C22×Dic3, C2×C3⋊D4, C6×D4, C3×C4○D4, D8⋊C22, S3×M4(2), C8.D6, D8⋊S3, D83S3, D4.D6, Q8.7D6, D126C22, Q8.14D6, C3×C8⋊C22, C2×D42S3, S3×C4○D4, D84D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, D8⋊C22, C2×S3×D4, D84D6

Smallest permutation representation of D84D6
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 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 31)(26 30)(27 29)(33 35)(36 40)(37 39)(41 47)(42 46)(43 45)
(1 46 32 21 34 14)(2 43 25 18 35 11)(3 48 26 23 36 16)(4 45 27 20 37 13)(5 42 28 17 38 10)(6 47 29 22 39 15)(7 44 30 19 40 12)(8 41 31 24 33 9)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 24A 24B order 1 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 24 24 size 1 1 2 4 4 4 6 6 12 2 2 2 3 3 4 6 12 12 12 2 4 8 8 8 4 4 12 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 D8⋊C22 D8⋊4D6 kernel D8⋊4D6 S3×M4(2) C8.D6 D8⋊S3 D8⋊3S3 D4.D6 Q8.7D6 D12⋊6C22 Q8.14D6 C3×C8⋊C22 C2×D4⋊2S3 S3×C4○D4 C8⋊C22 C4×S3 C2×Dic3 C22×S3 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 2 1 1 1 2 2 1 1 1 1 2 1

Matrix representation of D84D6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 72 0 0
,
 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 0 1 0 0 0 0 1 0
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 27 0 0 0 0 46 0 0 0 0 0 0 0 0 27 0 0 0 0 46 0
,
 1 72 0 0 0 0 0 72 0 0 0 0 0 0 0 27 0 0 0 0 46 0 0 0 0 0 0 0 0 46 0 0 0 0 27 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,1,0,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,46,0] >;`

D84D6 in GAP, Magma, Sage, TeX

`D_8\rtimes_4D_6`
`% in TeX`

`G:=Group("D8:4D6");`
`// GroupNames label`

`G:=SmallGroup(192,1332);`
`// by ID`

`G=gap.SmallGroup(192,1332);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,1123,185,438,235,102,6278]);`
`// 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,a*d=d*a,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

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