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## G = M4(2)⋊1F5order 320 = 26·5

### 1st 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)⋊1F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — C2×C4⋊F5 — M4(2)⋊1F5
 Lower central C5 — C10 — C20 — M4(2)⋊1F5
 Upper central C1 — C2 — C2×C4 — M4(2)

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

Subgroups: 490 in 118 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, 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, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C52C8, C40, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, M4(2)⋊C4, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C4×F5, C4⋊F5, C4⋊F5, C4⋊F5, C22⋊F5, C2×C4×D5, C22×F5, C40⋊C4, D5.D8, D5×M4(2), C2×C4⋊F5, D10.C23, M4(2)⋊1F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C8⋊C22, C8.C22, C2×F5, M4(2)⋊C4, C4⋊F5, C22×F5, C2×C4⋊F5, M4(2)⋊1F5

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4L 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 5 8 8 8 8 10 10 20 20 20 40 40 40 40 size 1 1 2 5 5 10 2 2 10 10 20 ··· 20 4 4 4 20 20 4 8 4 4 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + - - + + + - + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 Q8 Q8 D4 F5 C8⋊C22 C8.C22 C2×F5 C2×F5 C4⋊F5 C4⋊F5 M4(2)⋊1F5 kernel M4(2)⋊1F5 C40⋊C4 D5.D8 D5×M4(2) C2×C4⋊F5 D10.C23 C8⋊D5 C4.Dic5 C5×M4(2) C4×D5 C4×D5 C2×Dic5 C22×D5 M4(2) D5 D5 C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 1 1 1 1 1 1 1 2 1 2 2 2

Matrix representation of M4(2)⋊1F5 in GL8(𝔽41)

 7 0 14 14 0 0 0 0 27 34 27 0 0 0 0 0 0 27 34 27 0 0 0 0 14 14 0 7 0 0 0 0 0 0 0 0 5 34 36 0 0 0 0 0 17 0 0 36 0 0 0 0 22 9 36 7 0 0 0 0 9 1 24 0
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 39 11 1 0 0 0 0 0 26 0 0 1
,
 40 40 40 40 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 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
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 40 0 0 0 0 0 0 2 0 40 39 0 0 0 0 0 0 0 1

`G:=sub<GL(8,GF(41))| [7,27,0,14,0,0,0,0,0,34,27,14,0,0,0,0,14,27,34,0,0,0,0,0,14,0,27,7,0,0,0,0,0,0,0,0,5,17,22,9,0,0,0,0,34,0,9,1,0,0,0,0,36,0,36,24,0,0,0,0,0,36,7,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,39,26,0,0,0,0,0,40,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,40,2,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,39,1] >;`

M4(2)⋊1F5 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_1F_5`
`% in TeX`

`G:=Group("M4(2):1F5");`
`// GroupNames label`

`G:=SmallGroup(320,1065);`
`// by ID`

`G=gap.SmallGroup(320,1065);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,1684,102,6278,1595]);`
`// 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*c=c*b,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

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