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## G = M4(2).31D6order 192 = 26·3

### 4th non-split extension by M4(2) of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — M4(2).31D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C2×C4○D12 — M4(2).31D6
 Lower central C3 — C6 — C2×C6 — M4(2).31D6
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for M4(2).31D6
G = < a,b,c,d | a8=b2=1, c6=d2=a4, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 408 in 150 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2 [×5], C3, C4 [×4], C4 [×2], C22 [×3], C22 [×5], S3 [×2], C6, C6 [×3], C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C23 [×2], Dic3 [×2], C12 [×4], D6 [×4], C2×C6 [×3], C2×C6, C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], C3⋊C8 [×2], C24 [×2], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×6], C22×S3 [×2], C22×C6, C4.D4 [×2], C4.10D4 [×2], C2×M4(2), C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3 [×2], C4.Dic3, C2×C24, C3×M4(2) [×2], C3×M4(2), C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×4], C2×C3⋊D4 [×2], C22×C12, M4(2).8C22, C12.46D4 [×2], C12.47D4 [×2], C2×C4.Dic3, C6×M4(2), C2×C4○D12, M4(2).31D6
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], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, M4(2).8C22, C2×D6⋊C4, M4(2).31D6

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 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 3 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 2 2 2 12 12 2 1 1 2 2 2 12 12 2 2 2 4 4 4 4 4 4 12 12 12 12 2 2 2 2 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 S3 D4 D6 D6 C4×S3 D12 C3⋊D4 C4×S3 M4(2).8C22 M4(2).31D6 kernel M4(2).31D6 C12.46D4 C12.47D4 C2×C4.Dic3 C6×M4(2) C2×C4○D12 S3×C2×C4 C2×C3⋊D4 C2×M4(2) C2×C12 M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C23 C3 C1 # reps 1 2 2 1 1 1 4 4 1 4 2 1 2 4 4 2 2 4

Matrix representation of M4(2).31D6 in GL4(𝔽73) generated by

 0 0 46 0 0 0 0 46 43 60 0 0 13 30 0 0
,
 1 0 0 0 0 1 0 0 0 0 72 0 0 0 0 72
,
 0 27 0 0 46 27 0 0 0 0 0 27 0 0 46 27
,
 27 0 0 0 27 46 0 0 0 0 27 0 0 0 27 46
`G:=sub<GL(4,GF(73))| [0,0,43,13,0,0,60,30,46,0,0,0,0,46,0,0],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[0,46,0,0,27,27,0,0,0,0,0,46,0,0,27,27],[27,27,0,0,0,46,0,0,0,0,27,27,0,0,0,46] >;`

M4(2).31D6 in GAP, Magma, Sage, TeX

`M_4(2)._{31}D_6`
`% in TeX`

`G:=Group("M4(2).31D6");`
`// GroupNames label`

`G:=SmallGroup(192,691);`
`// by ID`

`G=gap.SmallGroup(192,691);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,422,58,1123,136,438,6278]);`
`// Polycyclic`

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

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