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## G = M4(2).19D6order 192 = 26·3

### 2nd non-split extension by M4(2) of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — M4(2).19D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — C2×D4⋊2S3 — M4(2).19D6
 Lower central C3 — C6 — C2×C6 — M4(2).19D6
 Upper central C1 — C2 — C2×C4 — C4.D4

Generators and relations for M4(2).19D6
G = < a,b,c,d | a8=b2=c6=1, d2=a4, bab=a5, cac-1=ab, dad-1=a5b, bc=cb, bd=db, dcd-1=a4c-1 >

Subgroups: 400 in 150 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×4], C22, C22 [×7], S3 [×2], C6, C6 [×3], C8 [×4], C2×C4, C2×C4 [×11], D4 [×6], Q8 [×2], C23 [×2], C23, Dic3 [×2], Dic3 [×2], C12 [×2], D6 [×2], D6, C2×C6, C2×C6 [×4], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C3⋊C8 [×2], C24 [×2], Dic6 [×2], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3, C22×C6 [×2], C4.D4, C4.D4, C4.10D4 [×2], C2×M4(2) [×2], C2×C4○D4, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3 [×2], C3×M4(2) [×2], C2×Dic6, S3×C2×C4, D42S3 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, M4(2).8C22, C12.47D4 [×2], C12.D4, C3×C4.D4, S3×M4(2) [×2], C2×D42S3, M4(2).19D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C2×C22⋊C4, S3×C2×C4, S3×D4 [×2], M4(2).8C22, S3×C22⋊C4, M4(2).19D6

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 8 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 S3 D4 D6 D6 C4×S3 S3×D4 M4(2).8C22 M4(2).19D6 kernel M4(2).19D6 C12.47D4 C12.D4 C3×C4.D4 S3×M4(2) C2×D4⋊2S3 C22×Dic3 C2×C3⋊D4 C4.D4 C4×S3 M4(2) C2×D4 C23 C4 C3 C1 # reps 1 2 1 1 2 1 4 4 1 4 2 1 4 2 2 1

Matrix representation of M4(2).19D6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 72 46 71 0 0 0 0 46 0 0 0 0 1 0 0 0 0 59 0 13 27
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 46 0 46 72
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 46 72 46 71 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 72 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 27 1 27 2 0 0 1 0 0 0 0 0 46 72 0 72

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,59,0,0,72,0,1,0,0,0,46,46,0,13,0,0,71,0,0,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,46,0,0,0,72,0,0,0,0,0,0,1,46,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,46,1,0,0,0,0,72,0,0,0,0,1,46,0,0,0,0,0,71,0,1],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,27,1,46,0,0,0,1,0,72,0,0,72,27,0,0,0,0,0,2,0,72] >;`

M4(2).19D6 in GAP, Magma, Sage, TeX

`M_4(2)._{19}D_6`
`% in TeX`

`G:=Group("M4(2).19D6");`
`// GroupNames label`

`G:=SmallGroup(192,304);`
`// by ID`

`G=gap.SmallGroup(192,304);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,219,58,570,136,438,6278]);`
`// Polycyclic`

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

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