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## G = M4(2).22D6order 192 = 26·3

### 5th 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).22D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — Q8.13D6 — M4(2).22D6
 Lower central C3 — C6 — C12 — M4(2).22D6
 Upper central C1 — C4 — C2×C4 — C4≀C2

Generators and relations for M4(2).22D6
G = < a,b,c,d | a8=b2=c6=1, d2=a6b, bab=a5, cac-1=dad-1=a-1b, cbc-1=a4b, bd=db, dcd-1=a6bc-1 >

Subgroups: 240 in 104 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×3], C22, C22 [×2], S3, C6, C6 [×2], C8 [×6], C2×C4, C2×C4 [×3], D4, D4 [×3], Q8, Q8, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C2×C6, C42, C2×C8 [×4], M4(2), M4(2) [×3], D8, SD16 [×2], Q16, C4○D4, C4○D4, C3⋊C8 [×4], C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C8⋊C4, C4≀C2, C4≀C2, C8.C4, C8○D4 [×2], C4○D8, S3×C8, C8⋊S3, C2×C3⋊C8 [×2], C2×C3⋊C8, C4.Dic3, C4.Dic3, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C4×C12, C3×M4(2), C4○D12, C3×C4○D4, C8.26D4, C42.S3, C424S3, C12.53D4, C3×C4≀C2, D12.C4, D4.Dic3, Q8.13D6, M4(2).22D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C22×S3, C4×D4, S3×C2×C4, S3×D4, D42S3, C8.26D4, Dic34D4, M4(2).22D6

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D4 D6 D6 D6 C4○D4 C4×S3 C4×S3 S3×D4 D4⋊2S3 C8.26D4 M4(2).22D6 kernel M4(2).22D6 C42.S3 C42⋊4S3 C12.53D4 C3×C4≀C2 D12.C4 D4.Dic3 Q8.13D6 D4⋊S3 D4.S3 Q8⋊2S3 C3⋊Q16 C4≀C2 C3⋊C8 C42 M4(2) C4○D4 C2×C6 D4 Q8 C4 C22 C3 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 1 2 1 1 1 2 2 2 1 1 2 4

Matrix representation of M4(2).22D6 in GL8(𝔽73)

 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 46 0 0 0 0 0 0 0 12 0 0 1 0 0 0 0 0 12 27 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72
,
 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 46 0 0 0 0 0 59 27 0
,
 71 18 0 0 0 0 0 0 20 2 0 0 0 0 0 0 0 0 2 55 0 0 0 0 0 0 53 71 0 0 0 0 0 0 0 0 39 0 0 3 0 0 0 0 0 8 3 0 0 0 0 0 0 67 65 0 0 0 0 0 13 0 0 34

`G:=sub<GL(8,GF(73))| [0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,46,12,0,0,0,0,0,72,0,0,12,0,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72],[0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,1,0,0,59,0,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0],[71,20,0,0,0,0,0,0,18,2,0,0,0,0,0,0,0,0,2,53,0,0,0,0,0,0,55,71,0,0,0,0,0,0,0,0,39,0,0,13,0,0,0,0,0,8,67,0,0,0,0,0,0,3,65,0,0,0,0,0,3,0,0,34] >;`

M4(2).22D6 in GAP, Magma, Sage, TeX

`M_4(2)._{22}D_6`
`% in TeX`

`G:=Group("M4(2).22D6");`
`// GroupNames label`

`G:=SmallGroup(192,382);`
`// by ID`

`G=gap.SmallGroup(192,382);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,219,58,136,1684,851,438,102,6278]);`
`// Polycyclic`

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

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