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G = M4(2).1F5order 320 = 26·5

1st non-split extension by M4(2) of 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 — Dic5 — 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=1, d4=a4, bab=a5, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 346 in 102 conjugacy classes, 48 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C8.C4, C2×M4(2), C52C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, M4(2).C4, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), D5⋊C8, C4.F5, C4.F5, C4.F5, C2×C5⋊C8, C22.F5, C2×C4×D5, C40.C4, D10.Q8, D5×M4(2), C2×C4.F5, D5⋊M4(2), 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, C2×F5, M4(2).C4, C4⋊F5, C22×F5, C2×C4⋊F5, M4(2).1F5

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5 8A 8B 8C ··· 8L 10A 10B 20A 20B 20C 40A 40B 40C 40D order 1 2 2 2 2 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 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 8 type + + + + + + + - + - + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 Q8 D4 Q8 F5 C2×F5 C2×F5 M4(2).C4 C4⋊F5 C4⋊F5 M4(2).1F5 kernel M4(2).1F5 C40.C4 D10.Q8 D5×M4(2) C2×C4.F5 D5⋊M4(2) C8⋊D5 C4.Dic5 C5×M4(2) C4×D5 C4×D5 C2×Dic5 C22×D5 M4(2) C8 C2×C4 C5 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 1 1 1 1 1 2 1 2 2 2 2

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

 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 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 1 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 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 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
,
 0 32 23 32 0 0 0 0 32 0 9 9 0 0 0 0 9 9 0 32 0 0 0 0 32 23 32 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 9 0 0 0 0 0 0 0 0 9 0 0

`G:=sub<GL(8,GF(41))| [34,14,0,27,0,0,0,0,0,7,14,27,0,0,0,0,27,14,7,0,0,0,0,0,27,0,14,34,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40],[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],[0,32,9,32,0,0,0,0,32,0,9,23,0,0,0,0,23,9,0,32,0,0,0,0,32,9,32,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

M4(2).1F5 in GAP, Magma, Sage, TeX

`M_4(2)._1F_5`
`% in TeX`

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

`G:=SmallGroup(320,1067);`
`// by ID`

`G=gap.SmallGroup(320,1067);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,136,1684,102,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^5=1,d^4=a^4,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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