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## G = M4(2)⋊3F5order 320 = 26·5

### 3rd semidirect product of M4(2) and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — M4(2)⋊3F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C2×C4×D5 — D10.C23 — M4(2)⋊3F5
 Lower central C5 — C10 — C20 — M4(2)⋊3F5
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for M4(2)⋊3F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a5, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=c3 >

Subgroups: 466 in 110 conjugacy classes, 36 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C42, C42⋊C2, C2×M4(2), C52C8, C40, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C426C4, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C4×F5, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C22×F5, D5×M4(2), C2×C4×F5, D10.C23, M4(2)⋊3F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C4≀C2, C2×F5, C426C4, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, M4(2)⋊3F5

Smallest permutation representation of M4(2)⋊3F5
On 40 points
Generators in S40
```(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)
(1 5)(3 7)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)
(1 26 22 16 35)(2 27 23 9 36)(3 28 24 10 37)(4 29 17 11 38)(5 30 18 12 39)(6 31 19 13 40)(7 32 20 14 33)(8 25 21 15 34)
(1 8)(2 3)(4 5)(6 7)(9 28 23 37)(10 27 24 36)(11 30 17 39)(12 29 18 38)(13 32 19 33)(14 31 20 40)(15 26 21 35)(16 25 22 34)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F ··· 4N 4O 4P 4Q 4R 5 8A 8B 8C 8D 10A 10B 20A 20B 20C 40A 40B 40C 40D order 1 2 2 2 2 2 4 4 4 4 4 4 ··· 4 4 4 4 4 5 8 8 8 8 10 10 20 20 20 40 40 40 40 size 1 1 2 5 5 10 1 1 2 5 5 10 ··· 10 20 20 20 20 4 4 4 20 20 4 8 4 4 8 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 8 type + + + + + - + + + + image C1 C2 C2 C2 C4 C4 C4 C4 D4 Q8 D4 C4≀C2 F5 C2×F5 C4×F5 C22⋊F5 C4⋊F5 M4(2)⋊3F5 kernel M4(2)⋊3F5 D5×M4(2) C2×C4×F5 D10.C23 C4.Dic5 C5×M4(2) C4×F5 C4⋊F5 C4×D5 C2×Dic5 C22×D5 D5 M4(2) C2×C4 C4 C4 C22 C1 # reps 1 1 1 1 2 2 4 4 2 1 1 8 1 1 2 2 2 2

Matrix representation of M4(2)⋊3F5 in GL6(𝔽41)

 28 37 0 0 0 0 40 13 0 0 0 0 0 0 7 27 0 14 0 0 0 34 27 14 0 0 14 27 34 0 0 0 14 0 27 7
,
 1 0 0 0 0 0 14 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 13 4 0 0 0 0 19 28 0 0 0 0 0 0 34 0 7 14 0 0 0 14 34 7 0 0 27 7 34 14 0 0 27 14 7 0

`G:=sub<GL(6,GF(41))| [28,40,0,0,0,0,37,13,0,0,0,0,0,0,7,0,14,14,0,0,27,34,27,0,0,0,0,27,34,27,0,0,14,14,0,7],[1,14,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[13,19,0,0,0,0,4,28,0,0,0,0,0,0,34,0,27,27,0,0,0,14,7,14,0,0,7,34,34,7,0,0,14,7,14,0] >;`

M4(2)⋊3F5 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_3F_5`
`% in TeX`

`G:=Group("M4(2):3F5");`
`// GroupNames label`

`G:=SmallGroup(320,238);`
`// by ID`

`G=gap.SmallGroup(320,238);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,136,1684,851,102,6278,3156]);`
`// Polycyclic`

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

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