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## G = M4(2).13D6order 192 = 26·3

### 13rd non-split extension by M4(2) of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — M4(2).13D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — Q8.14D6 — M4(2).13D6
 Lower central C3 — C6 — C2×C12 — M4(2).13D6
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for M4(2).13D6
G = < a,b,c,d | a8=b2=c6=1, d2=a2, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 272 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, D4.S3, C3⋊Q16, C3×M4(2), C3×D8, C3×SD16, C2×Dic6, C6×D4, C3×C4○D4, D4.3D4, C12.53D4, C12.47D4, C12.D4, C2×D4.S3, D4.Dic3, Q8.14D6, C3×C8⋊C22, M4(2).13D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S3×D4, D42S3, C2×C3⋊D4, D4.3D4, C23.14D6, M4(2).13D6

Character table of M4(2).13D6

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E 8F 8G 12A 12B 12C 24A 24B size 1 1 2 4 8 2 2 2 4 24 2 4 8 8 8 6 6 8 12 12 12 24 4 4 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 1 1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 -1 1 1 1 1 -1 1 1 1 -1 -1 1 1 -1 1 1 1 -1 1 1 1 -1 -1 linear of order 2 ρ4 1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 -1 1 1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 1 linear of order 2 ρ6 1 1 1 -1 1 1 1 1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ7 1 1 1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 1 -1 -1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 1 1 linear of order 2 ρ9 2 2 2 2 2 -1 2 2 2 0 -1 -1 -1 -1 -1 0 0 2 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 0 0 2 -2 2 0 0 2 -2 0 0 0 2 2 0 0 0 -2 0 -2 2 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 -2 -2 -1 2 2 -2 0 -1 -1 1 1 1 0 0 2 0 0 0 0 -1 -1 1 -1 -1 orthogonal lifted from D6 ρ12 2 2 -2 0 0 2 -2 2 0 0 2 -2 0 0 0 -2 -2 0 0 0 2 0 -2 2 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 -2 0 2 2 -2 2 0 2 -2 -2 0 0 0 0 0 0 0 0 0 2 -2 2 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -1 2 2 2 0 -1 -1 -1 1 1 0 0 -2 0 0 0 0 -1 -1 -1 1 1 orthogonal lifted from D6 ρ15 2 2 -2 2 0 2 2 -2 -2 0 2 -2 2 0 0 0 0 0 0 0 0 0 2 -2 -2 0 0 orthogonal lifted from D4 ρ16 2 2 2 -2 2 -1 2 2 -2 0 -1 -1 1 -1 -1 0 0 -2 0 0 0 0 -1 -1 1 1 1 orthogonal lifted from D6 ρ17 2 2 -2 -2 0 -1 2 -2 2 0 -1 1 1 -√-3 √-3 0 0 0 0 0 0 0 -1 1 -1 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 -2 2 0 -1 2 -2 -2 0 -1 1 -1 √-3 -√-3 0 0 0 0 0 0 0 -1 1 1 √-3 -√-3 complex lifted from C3⋊D4 ρ19 2 2 -2 -2 0 -1 2 -2 2 0 -1 1 1 √-3 -√-3 0 0 0 0 0 0 0 -1 1 -1 -√-3 √-3 complex lifted from C3⋊D4 ρ20 2 2 -2 2 0 -1 2 -2 -2 0 -1 1 -1 -√-3 √-3 0 0 0 0 0 0 0 -1 1 1 -√-3 √-3 complex lifted from C3⋊D4 ρ21 2 2 2 0 0 2 -2 -2 0 0 2 2 0 0 0 0 0 0 -2i 2i 0 0 -2 -2 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 0 0 2 -2 -2 0 0 2 2 0 0 0 0 0 0 2i -2i 0 0 -2 -2 0 0 0 complex lifted from C4○D4 ρ23 4 4 -4 0 0 -2 -4 4 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 2 -2 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 4 0 0 -2 -4 -4 0 0 -2 -2 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ25 4 -4 0 0 0 4 0 0 0 0 -4 0 0 0 0 -2√-2 2√-2 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.3D4 ρ26 4 -4 0 0 0 4 0 0 0 0 -4 0 0 0 0 2√-2 -2√-2 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.3D4 ρ27 8 -8 0 0 0 -4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of M4(2).13D6 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 0 72 0 0 0 0 0 72 72 72 48 0 0 0 0 0 72 0 0 0 0 0 0 3 3 3 1
,
 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 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 3 3 0 1
,
 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 72 72 72 48 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 55 71 0 0 0 0 0 0 53 18 0 0 0 0 0 0 0 0 18 2 0 0 0 0 0 0 20 55 0 0 0 0 0 0 0 0 6 6 12 4 0 0 0 0 6 6 0 4 0 0 0 0 67 67 0 0 0 0 0 0 0 55 55 61

`G:=sub<GL(8,GF(73))| [0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,3,0,0,0,0,0,72,72,3,0,0,0,0,72,72,0,3,0,0,0,0,0,48,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,3,0,0,0,0,0,72,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,48,0,1],[55,53,0,0,0,0,0,0,71,18,0,0,0,0,0,0,0,0,18,20,0,0,0,0,0,0,2,55,0,0,0,0,0,0,0,0,6,6,67,0,0,0,0,0,6,6,67,55,0,0,0,0,12,0,0,55,0,0,0,0,4,4,0,61] >;`

M4(2).13D6 in GAP, Magma, Sage, TeX

`M_4(2)._{13}D_6`
`% in TeX`

`G:=Group("M4(2).13D6");`
`// GroupNames label`

`G:=SmallGroup(192,759);`
`// by ID`

`G=gap.SmallGroup(192,759);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,219,1123,297,136,1684,851,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^2,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;`
`// generators/relations`

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