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## G = D6⋊6M4(2)  order 192 = 26·3

### 2nd semidirect product of D6 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D6⋊6M4(2)
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — S3×C22×C4 — D6⋊6M4(2)
 Lower central C3 — C2×C6 — D6⋊6M4(2)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for D66M4(2)
G = < a,b,c,d | a6=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=c5 >

Subgroups: 504 in 190 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22⋊C8, C2×M4(2), C2×M4(2), C23×C4, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C24.4C4, D6⋊C8, C2×C4.Dic3, C6×M4(2), S3×C22×C4, D66M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, M4(2), C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C2×M4(2), D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C24.4C4, S3×M4(2), C2×D6⋊C4, D66M4(2)

Smallest permutation representation of D66M4(2)
On 48 points
Generators in S48
```(1 25 47 16 23 36)(2 26 48 9 24 37)(3 27 41 10 17 38)(4 28 42 11 18 39)(5 29 43 12 19 40)(6 30 44 13 20 33)(7 31 45 14 21 34)(8 32 46 15 22 35)
(1 36)(2 48)(3 38)(4 42)(5 40)(6 44)(7 34)(8 46)(9 37)(10 41)(11 39)(12 43)(13 33)(14 45)(15 35)(16 47)(17 27)(19 29)(21 31)(23 25)
(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 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)```

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D4 D6 D6 M4(2) C4×S3 D12 C3⋊D4 C4×S3 S3×M4(2) kernel D6⋊6M4(2) D6⋊C8 C2×C4.Dic3 C6×M4(2) S3×C22×C4 S3×C2×C4 C22×Dic3 S3×C23 C2×M4(2) C2×C12 C2×C8 C22×C4 D6 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 4 2 2 1 4 2 1 8 2 4 4 2 4

Matrix representation of D66M4(2) in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 0 1 0 0 72 1
,
 72 0 0 0 0 1 0 0 0 0 72 1 0 0 0 1
,
 0 27 0 0 72 0 0 0 0 0 30 13 0 0 60 43
,
 72 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
`G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,1],[72,0,0,0,0,1,0,0,0,0,72,0,0,0,1,1],[0,72,0,0,27,0,0,0,0,0,30,60,0,0,13,43],[72,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1] >;`

D66M4(2) in GAP, Magma, Sage, TeX

`D_6\rtimes_6M_4(2)`
`% in TeX`

`G:=Group("D6:6M4(2)");`
`// GroupNames label`

`G:=SmallGroup(192,685);`
`// by ID`

`G=gap.SmallGroup(192,685);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,136,6278]);`
`// Polycyclic`

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

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