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## G = M4(2)⋊D6order 192 = 26·3

### 1st semidirect product of M4(2) and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — M4(2)⋊D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — D4⋊6D6 — M4(2)⋊D6
 Lower central C3 — C6 — C2×C12 — M4(2)⋊D6
 Upper central C1 — C2 — C2×C4 — C4.D4

Generators and relations for M4(2)⋊D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=a5, cac-1=ab, dad=a-1b, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 528 in 152 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C24⋊C2, Dic12, C4×Dic3, C6.D4, C3×M4(2), C2×Dic6, C4○D12, S3×D4, D42S3, C2×C3⋊D4, C6×D4, D4.9D4, D12⋊C4, C3×C4.D4, C8.D6, C23.12D6, D46D6, M4(2)⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C2×D12, S3×D4, D4.9D4, D6⋊D4, M4(2)⋊D6

Character table of M4(2)⋊D6

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 6D 8A 8B 12A 12B 24A 24B 24C 24D size 1 1 2 4 4 12 12 2 2 2 12 12 12 12 24 2 4 8 8 8 8 4 4 8 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 0 0 -2 0 2 -2 2 0 0 2 0 0 2 -2 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 0 0 2 2 2 -2 0 -2 0 0 0 2 -2 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 0 0 -1 2 2 0 0 0 0 0 -1 -1 -1 -1 -2 -2 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ12 2 2 2 -2 -2 0 0 -1 2 2 0 0 0 0 0 -1 -1 1 1 -2 2 -1 -1 -1 1 -1 1 orthogonal lifted from D6 ρ13 2 2 -2 0 0 0 -2 2 2 -2 0 2 0 0 0 2 -2 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 0 2 0 2 -2 2 0 0 -2 0 0 2 -2 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 2 0 0 -1 2 2 0 0 0 0 0 -1 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ16 2 2 2 -2 -2 0 0 -1 2 2 0 0 0 0 0 -1 -1 1 1 2 -2 -1 -1 1 -1 1 -1 orthogonal lifted from D6 ρ17 2 2 2 2 -2 0 0 2 -2 -2 0 0 0 0 0 2 2 -2 2 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 2 0 0 2 -2 -2 0 0 0 0 0 2 2 2 -2 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 0 0 -1 -2 -2 0 0 0 0 0 -1 -1 1 -1 0 0 1 1 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ20 2 2 2 -2 2 0 0 -1 -2 -2 0 0 0 0 0 -1 -1 -1 1 0 0 1 1 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ21 2 2 2 -2 2 0 0 -1 -2 -2 0 0 0 0 0 -1 -1 -1 1 0 0 1 1 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ22 2 2 2 2 -2 0 0 -1 -2 -2 0 0 0 0 0 -1 -1 1 -1 0 0 1 1 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ23 4 4 -4 0 0 0 0 -2 -4 4 0 0 0 0 0 -2 2 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 -4 0 0 0 0 -2 4 -4 0 0 0 0 0 -2 2 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 0 0 0 4 0 0 2i 0 0 -2i 0 -4 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.9D4 ρ26 4 -4 0 0 0 0 0 4 0 0 -2i 0 0 2i 0 -4 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D4.9D4 ρ27 8 -8 0 0 0 0 0 -4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of M4(2)⋊D6 in GL8(𝔽73)

 59 0 23 23 0 0 0 0 0 59 27 23 0 0 0 0 67 6 14 0 0 0 0 0 61 67 0 14 0 0 0 0 0 0 0 0 0 46 64 9 0 0 0 0 27 0 64 9 0 0 0 0 4 69 46 0 0 0 0 0 4 69 0 27
,
 72 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 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 59 66 0 72 0 0 0 0 66 59 1 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 72 0 0 0 0 3 0 72 0
,
 1 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 66 59 1 72 0 0 0 0 59 66 0 72 0 0 0 0 0 0 0 0 0 72 0 25 0 0 0 0 1 0 48 0 0 0 0 0 0 70 0 1 0 0 0 0 3 0 72 0

`G:=sub<GL(8,GF(73))| [59,0,67,61,0,0,0,0,0,59,6,67,0,0,0,0,23,27,14,0,0,0,0,0,23,23,0,14,0,0,0,0,0,0,0,0,0,27,4,4,0,0,0,0,46,0,69,69,0,0,0,0,64,64,46,0,0,0,0,0,9,9,0,27],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,72,59,66,0,0,0,0,1,0,66,59,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,1,0,3,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0],[1,0,66,59,0,0,0,0,1,72,59,66,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,72,0,70,0,0,0,0,0,0,48,0,72,0,0,0,0,25,0,1,0] >;`

M4(2)⋊D6 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes D_6`
`% in TeX`

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

`G:=SmallGroup(192,305);`
`// by ID`

`G=gap.SmallGroup(192,305);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,58,1123,570,136,1684,438,6278]);`
`// Polycyclic`

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

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