Copied to
clipboard

## G = M4(2).F5order 320 = 26·5

### 3rd 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).F5
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C2×C4×D5 — D5⋊M4(2) — M4(2).F5
 Lower central C5 — C10 — C20 — M4(2).F5
 Upper central C1 — C4 — C2×C4 — M4(2)

Generators and relations for M4(2).F5
G = < a,b,c,d | a8=b2=c5=1, d4=a4, bab=a5, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a4b, dcd-1=c3 >

Subgroups: 322 in 90 conjugacy classes, 36 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, D5, C10, C10, C2×C8, M4(2), M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C22×C8, C2×M4(2), C52C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C4.C42, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), D5⋊C8, D5⋊C8, C4.F5, C2×C5⋊C8, C22.F5, C2×C4×D5, D5×M4(2), C2×D5⋊C8, D5⋊M4(2), M4(2).F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C8.C4, C2×F5, C4.C42, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, M4(2).F5

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5 8A 8B 8C ··· 8J 8K ··· 8P 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 8 ··· 8 10 10 20 20 20 40 40 40 40 size 1 1 2 5 5 10 1 1 2 5 5 10 4 4 4 10 ··· 10 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 D4 Q8 C8.C4 F5 C2×F5 C4×F5 C22⋊F5 C4⋊F5 M4(2).F5 kernel M4(2).F5 D5×M4(2) C2×D5⋊C8 D5⋊M4(2) C4.Dic5 C5×M4(2) D5⋊C8 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).F5 in GL6(𝔽41)

 28 4 0 0 0 0 17 13 0 0 0 0 0 0 34 14 0 27 0 0 0 7 14 27 0 0 27 14 7 0 0 0 27 0 14 34
,
 40 0 0 0 0 0 14 1 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
,
 39 29 0 0 0 0 3 2 0 0 0 0 0 0 37 0 4 10 0 0 0 10 35 6 0 0 31 6 35 10 0 0 31 10 4 0

`G:=sub<GL(6,GF(41))| [28,17,0,0,0,0,4,13,0,0,0,0,0,0,34,0,27,27,0,0,14,7,14,0,0,0,0,14,7,14,0,0,27,27,0,34],[40,14,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[39,3,0,0,0,0,29,2,0,0,0,0,0,0,37,0,31,31,0,0,0,10,6,10,0,0,4,35,35,4,0,0,10,6,10,0] >;`

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

`M_4(2).F_5`
`% in TeX`

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

`G:=SmallGroup(320,239);`
`// by ID`

`G=gap.SmallGroup(320,239);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,136,1684,851,102,6278,3156]);`
`// 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,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;`
`// generators/relations`

׿
×
𝔽