Copied to
clipboard

## G = M4(2)⋊F5order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 418 in 98 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C4⋊C4, C2×M4(2), C52C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C22.C42, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), D5⋊C8, C4.F5, C4⋊F5, C22.F5, C2×C4×D5, C22×F5, D5×M4(2), D5⋊M4(2), C2×C4⋊F5, M4(2)⋊F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C4.D4, C4.10D4, C2×F5, C22.C42, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, M4(2)⋊F5

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5 8A 8B 8C ··· 8H 10A 10B 20A 20B 20C 40A 40B 40C 40D order 1 2 2 2 2 2 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 2 2 10 10 20 20 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 2 2 4 4 4 4 4 4 4 8 type + + + + + - + + - + + image C1 C2 C2 C2 C4 C4 C4 C4 D4 Q8 F5 C4.D4 C4.10D4 C2×F5 C4⋊F5 C22⋊F5 C4×F5 M4(2)⋊F5 kernel M4(2)⋊F5 D5×M4(2) D5⋊M4(2) C2×C4⋊F5 C4.Dic5 C5×M4(2) C22.F5 C22×F5 C4×D5 C4×D5 M4(2) D5 D5 C2×C4 C4 C4 C22 C1 # reps 1 1 1 1 2 2 4 4 3 1 1 1 1 1 2 2 2 2

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

 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 36 4 39 0 0 0 0 0 4 36 0 39 0 0 0 0 0 1 5 37 0 0 0 0 0 0 37 5
,
 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 4 40 0 0 0 0 0 4 36 0 40
,
 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 0 40 0 0 0 0 0 0 0 37 1 0 0 0 0 0 4 0 0 40

`G:=sub<GL(8,GF(41))| [9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,36,4,0,0,0,0,0,0,4,36,1,0,0,0,0,0,39,0,5,37,0,0,0,0,0,39,37,5],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,36,4,0,0,0,0,0,1,4,36,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[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],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,0,0,4,0,0,0,0,0,40,37,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40] >;`

M4(2)⋊F5 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes F_5`
`% in TeX`

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

`G:=SmallGroup(320,237);`
`// by ID`

`G=gap.SmallGroup(320,237);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1123,136,851,6278,3156]);`
`// 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*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

׿
×
𝔽