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## G = M4(2).25D6order 192 = 26·3

### 8th non-split extension by M4(2) of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — M4(2).25D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — C2×C8⋊S3 — M4(2).25D6
 Lower central C3 — C6 — C12 — M4(2).25D6
 Upper central C1 — C4 — C2×C4 — C8.C4

Generators and relations for M4(2).25D6
G = < a,b,c,d | a8=b2=1, c6=d2=a6b, bab=a5, cac-1=a-1b, dad-1=a3b, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 224 in 102 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], S3 [×2], C6, C6, C8 [×2], C8 [×6], C2×C4, C2×C4 [×5], C23, Dic3 [×2], C12 [×2], D6 [×2], D6, C2×C6, C2×C8, C2×C8 [×3], M4(2) [×2], M4(2) [×8], C22×C4, C3⋊C8 [×2], C3⋊C8 [×2], C24 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C8.C4, C8.C4 [×3], C2×M4(2) [×3], S3×C8 [×2], C8⋊S3 [×4], C8⋊S3 [×2], C2×C3⋊C8, C4.Dic3 [×2], C2×C24, C3×M4(2) [×2], S3×C2×C4, M4(2).C4, C24.C4, C12.53D4 [×2], C3×C8.C4, C2×C8⋊S3, S3×M4(2) [×2], M4(2).25D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4×S3 [×2], C22×S3, C2×C4⋊C4, S3×C2×C4, S3×D4, S3×Q8, M4(2).C4, S3×C4⋊C4, M4(2).25D6

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - - + + + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 Q8 Q8 D6 D6 C4×S3 S3×D4 S3×Q8 M4(2).C4 M4(2).25D6 kernel M4(2).25D6 C24.C4 C12.53D4 C3×C8.C4 C2×C8⋊S3 S3×M4(2) C8⋊S3 C8.C4 C4×S3 C2×Dic3 C22×S3 C2×C8 M4(2) C8 C4 C22 C3 C1 # reps 1 1 2 1 1 2 8 1 2 1 1 1 2 4 1 1 2 4

Matrix representation of M4(2).25D6 in GL4(𝔽5) generated by

 0 0 3 3 0 0 1 2 2 2 0 0 4 3 0 0
,
 1 0 0 0 0 1 0 0 0 0 4 0 0 0 0 4
,
 2 4 0 0 4 4 0 0 0 0 3 2 0 0 2 4
,
 1 4 0 0 4 4 0 0 0 0 1 2 0 0 1 4
`G:=sub<GL(4,GF(5))| [0,0,2,4,0,0,2,3,3,1,0,0,3,2,0,0],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[2,4,0,0,4,4,0,0,0,0,3,2,0,0,2,4],[1,4,0,0,4,4,0,0,0,0,1,1,0,0,2,4] >;`

M4(2).25D6 in GAP, Magma, Sage, TeX

`M_4(2)._{25}D_6`
`% in TeX`

`G:=Group("M4(2).25D6");`
`// GroupNames label`

`G:=SmallGroup(192,452);`
`// by ID`

`G=gap.SmallGroup(192,452);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,219,58,136,438,102,6278]);`
`// Polycyclic`

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

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