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## G = D6⋊M4(2)  order 192 = 26·3

### 1st 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⋊M4(2)
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — S3×C22×C4 — D6⋊M4(2)
 Lower central C3 — C2×C6 — D6⋊M4(2)
 Upper central C1 — C2×C4 — C22⋊C8

Generators and relations for D6⋊M4(2)
G = < a,b,c,d | a6=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a4b, dcd=c5 >

Subgroups: 512 in 190 conjugacy classes, 61 normal (33 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×2], C4 [×4], C22, C22 [×2], C22 [×18], S3 [×4], C6 [×3], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×16], C23, C23 [×8], Dic3 [×2], Dic3, C12 [×2], C12, D6 [×4], D6 [×12], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×9], C24, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C4×S3 [×6], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×2], C22×S3 [×2], C22×S3 [×6], C22×C6, C22⋊C8, C22⋊C8 [×3], C2×M4(2) [×2], C23×C4, C8⋊S3 [×4], C2×C3⋊C8 [×2], C2×C24 [×2], S3×C2×C4 [×4], S3×C2×C4 [×4], C22×Dic3, C22×C12, S3×C23, C24.4C4, D6⋊C8 [×2], C12.55D4, C3×C22⋊C8, C2×C8⋊S3 [×2], S3×C22×C4, D6⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C2×C22⋊C4, C2×M4(2) [×2], C8⋊S3 [×2], S3×C2×C4, S3×D4 [×2], C24.4C4, S3×C22⋊C4, C2×C8⋊S3, S3×M4(2), D6⋊M4(2)

Smallest permutation representation of D6⋊M4(2)
On 48 points
Generators in S48
```(1 22 32 15 46 40)(2 33 47 16 25 23)(3 24 26 9 48 34)(4 35 41 10 27 17)(5 18 28 11 42 36)(6 37 43 12 29 19)(7 20 30 13 44 38)(8 39 45 14 31 21)
(1 32)(2 23)(3 26)(4 17)(5 28)(6 19)(7 30)(8 21)(9 34)(10 41)(11 36)(12 43)(13 38)(14 45)(15 40)(16 47)(25 33)(27 35)(29 37)(31 39)
(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 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 27)(18 32)(19 29)(20 26)(21 31)(22 28)(23 25)(24 30)(33 47)(34 44)(35 41)(36 46)(37 43)(38 48)(39 45)(40 42)```

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

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

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

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 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D4 D6 D6 M4(2) M4(2) C4×S3 C4×S3 C8⋊S3 S3×D4 S3×M4(2) kernel D6⋊M4(2) D6⋊C8 C12.55D4 C3×C22⋊C8 C2×C8⋊S3 S3×C22×C4 S3×C2×C4 C22×Dic3 S3×C23 C22⋊C8 C4×S3 C2×C8 C22×C4 D6 C2×C6 C2×C4 C23 C22 C4 C2 # reps 1 2 1 1 2 1 4 2 2 1 4 2 1 4 4 2 2 8 2 2

Matrix representation of D6⋊M4(2) in GL4(𝔽73) generated by

 0 1 0 0 72 1 0 0 0 0 72 0 0 0 0 72
,
 1 72 0 0 0 72 0 0 0 0 1 0 0 0 72 72
,
 16 65 0 0 8 57 0 0 0 0 1 2 0 0 0 72
,
 0 1 0 0 1 0 0 0 0 0 72 0 0 0 0 72
`G:=sub<GL(4,GF(73))| [0,72,0,0,1,1,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,72,72,0,0,0,0,1,72,0,0,0,72],[16,8,0,0,65,57,0,0,0,0,1,0,0,0,2,72],[0,1,0,0,1,0,0,0,0,0,72,0,0,0,0,72] >;`

D6⋊M4(2) in GAP, Magma, Sage, TeX

`D_6\rtimes M_4(2)`
`% in TeX`

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

`G:=SmallGroup(192,285);`
`// by ID`

`G=gap.SmallGroup(192,285);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,58,136,6278]);`
`// Polycyclic`

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

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