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## G = M4(2)⋊4F5order 320 = 26·5

### 4th 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)⋊4F5
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C2×C4×D5 — C2×C4.F5 — M4(2)⋊4F5
 Lower central C5 — C10 — C2×C10 — M4(2)⋊4F5
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for M4(2)⋊4F5
G = < a,b,c,d | a8=b2=c5=d4=1, bab=a5, ac=ca, dad-1=a5b, bc=cb, dbd-1=a4b, dcd-1=c3 >

Subgroups: 370 in 90 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C42⋊C2, C2×M4(2), C52C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, M4(2)⋊4C4, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C4.F5, C4×F5, C4⋊F5, C2×C5⋊C8, C22⋊F5, C2×C4×D5, D5×M4(2), C2×C4.F5, D10.C23, M4(2)⋊4F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C2×F5, M4(2)⋊4C4, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, M4(2)⋊4F5

Smallest permutation representation of M4(2)⋊4F5
On 80 points
Generators in S80
```(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 58)(18 63)(19 60)(20 57)(21 62)(22 59)(23 64)(24 61)(25 44)(26 41)(27 46)(28 43)(29 48)(30 45)(31 42)(32 47)(33 80)(34 77)(35 74)(36 79)(37 76)(38 73)(39 78)(40 75)(49 72)(50 69)(51 66)(52 71)(53 68)(54 65)(55 70)(56 67)
(1 20 42 40 66)(2 21 43 33 67)(3 22 44 34 68)(4 23 45 35 69)(5 24 46 36 70)(6 17 47 37 71)(7 18 48 38 72)(8 19 41 39 65)(9 61 27 79 55)(10 62 28 80 56)(11 63 29 73 49)(12 64 30 74 50)(13 57 31 75 51)(14 58 32 76 52)(15 59 25 77 53)(16 60 26 78 54)
(1 3)(2 12 6 16)(4 10 8 14)(5 7)(9 15)(11 13)(17 26 67 74)(18 46 72 36)(19 32 69 80)(20 44 66 34)(21 30 71 78)(22 42 68 40)(23 28 65 76)(24 48 70 38)(25 55 77 61)(27 53 79 59)(29 51 73 57)(31 49 75 63)(33 64 47 54)(35 62 41 52)(37 60 43 50)(39 58 45 56)```

`G:=sub<Sym(80)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)(17,58)(18,63)(19,60)(20,57)(21,62)(22,59)(23,64)(24,61)(25,44)(26,41)(27,46)(28,43)(29,48)(30,45)(31,42)(32,47)(33,80)(34,77)(35,74)(36,79)(37,76)(38,73)(39,78)(40,75)(49,72)(50,69)(51,66)(52,71)(53,68)(54,65)(55,70)(56,67), (1,20,42,40,66)(2,21,43,33,67)(3,22,44,34,68)(4,23,45,35,69)(5,24,46,36,70)(6,17,47,37,71)(7,18,48,38,72)(8,19,41,39,65)(9,61,27,79,55)(10,62,28,80,56)(11,63,29,73,49)(12,64,30,74,50)(13,57,31,75,51)(14,58,32,76,52)(15,59,25,77,53)(16,60,26,78,54), (1,3)(2,12,6,16)(4,10,8,14)(5,7)(9,15)(11,13)(17,26,67,74)(18,46,72,36)(19,32,69,80)(20,44,66,34)(21,30,71,78)(22,42,68,40)(23,28,65,76)(24,48,70,38)(25,55,77,61)(27,53,79,59)(29,51,73,57)(31,49,75,63)(33,64,47,54)(35,62,41,52)(37,60,43,50)(39,58,45,56)>;`

`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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)(17,58)(18,63)(19,60)(20,57)(21,62)(22,59)(23,64)(24,61)(25,44)(26,41)(27,46)(28,43)(29,48)(30,45)(31,42)(32,47)(33,80)(34,77)(35,74)(36,79)(37,76)(38,73)(39,78)(40,75)(49,72)(50,69)(51,66)(52,71)(53,68)(54,65)(55,70)(56,67), (1,20,42,40,66)(2,21,43,33,67)(3,22,44,34,68)(4,23,45,35,69)(5,24,46,36,70)(6,17,47,37,71)(7,18,48,38,72)(8,19,41,39,65)(9,61,27,79,55)(10,62,28,80,56)(11,63,29,73,49)(12,64,30,74,50)(13,57,31,75,51)(14,58,32,76,52)(15,59,25,77,53)(16,60,26,78,54), (1,3)(2,12,6,16)(4,10,8,14)(5,7)(9,15)(11,13)(17,26,67,74)(18,46,72,36)(19,32,69,80)(20,44,66,34)(21,30,71,78)(22,42,68,40)(23,28,65,76)(24,48,70,38)(25,55,77,61)(27,53,79,59)(29,51,73,57)(31,49,75,63)(33,64,47,54)(35,62,41,52)(37,60,43,50)(39,58,45,56) );`

`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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,13),(2,10),(3,15),(4,12),(5,9),(6,14),(7,11),(8,16),(17,58),(18,63),(19,60),(20,57),(21,62),(22,59),(23,64),(24,61),(25,44),(26,41),(27,46),(28,43),(29,48),(30,45),(31,42),(32,47),(33,80),(34,77),(35,74),(36,79),(37,76),(38,73),(39,78),(40,75),(49,72),(50,69),(51,66),(52,71),(53,68),(54,65),(55,70),(56,67)], [(1,20,42,40,66),(2,21,43,33,67),(3,22,44,34,68),(4,23,45,35,69),(5,24,46,36,70),(6,17,47,37,71),(7,18,48,38,72),(8,19,41,39,65),(9,61,27,79,55),(10,62,28,80,56),(11,63,29,73,49),(12,64,30,74,50),(13,57,31,75,51),(14,58,32,76,52),(15,59,25,77,53),(16,60,26,78,54)], [(1,3),(2,12,6,16),(4,10,8,14),(5,7),(9,15),(11,13),(17,26,67,74),(18,46,72,36),(19,32,69,80),(20,44,66,34),(21,30,71,78),(22,42,68,40),(23,28,65,76),(24,48,70,38),(25,55,77,61),(27,53,79,59),(29,51,73,57),(31,49,75,63),(33,64,47,54),(35,62,41,52),(37,60,43,50),(39,58,45,56)]])`

32 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 5 8A 8B 8C ··· 8H 10A 10B 20A 20B 20C 40A 40B 40C 40D order 1 2 2 2 2 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 10 10 2 2 5 5 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 8 type + + + + + - + + + image C1 C2 C2 C2 C4 C4 C4 C4 D4 Q8 F5 C2×F5 M4(2)⋊4C4 C4⋊F5 C22⋊F5 C4×F5 M4(2)⋊4F5 kernel M4(2)⋊4F5 D5×M4(2) C2×C4.F5 D10.C23 C4.Dic5 C5×M4(2) C2×C5⋊C8 C22⋊F5 C4×D5 C4×D5 M4(2) C2×C4 C5 C4 C4 C22 C1 # reps 1 1 1 1 2 2 4 4 3 1 1 1 2 2 2 2 2

Matrix representation of M4(2)⋊4F5 in GL8(𝔽41)

 40 0 4 4 0 0 0 0 0 0 1 0 0 0 0 0 18 9 1 1 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 34 0 27 27 0 0 0 0 14 7 14 0 0 0 0 0 0 14 7 14 0 0 0 0 27 27 0 34
,
 1 0 32 32 0 0 0 0 39 40 9 9 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 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
,
 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 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
,
 32 32 40 0 0 0 0 0 18 9 1 1 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 0 1 1 1 1

`G:=sub<GL(8,GF(41))| [40,0,18,0,0,0,0,0,0,0,9,32,0,0,0,0,4,1,1,0,0,0,0,0,4,0,1,0,0,0,0,0,0,0,0,0,34,14,0,27,0,0,0,0,0,7,14,27,0,0,0,0,27,14,7,0,0,0,0,0,27,0,14,34],[1,39,0,0,0,0,0,0,0,40,0,0,0,0,0,0,32,9,0,40,0,0,0,0,32,9,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[32,18,0,0,0,0,0,0,32,9,0,0,0,0,0,0,40,1,9,0,0,0,0,0,0,1,0,32,0,0,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,1,0,0,0,0,0,40,0,1] >;`

M4(2)⋊4F5 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes_4F_5`
`% in TeX`

`G:=Group("M4(2):4F5");`
`// GroupNames label`

`G:=SmallGroup(320,240);`
`// by ID`

`G=gap.SmallGroup(320,240);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1123,136,851,6278,3156]);`
`// 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^5*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;`
`// generators/relations`

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