Copied to
clipboard

## G = C52⋊8M4(2)  order 400 = 24·52

### 4th semidirect product of C52 and M4(2) acting via M4(2)/C4=C4

Aliases: C20.4F5, C528M4(2), (C5×C20).4C4, C4.(C52⋊C4), C52(C4.F5), C525C85C2, C10.22(C2×F5), C526C4.23C22, (C4×C5⋊D5).8C2, (C2×C5⋊D5).10C4, C2.4(C2×C52⋊C4), (C5×C10).35(C2×C4), SmallGroup(400,157)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C52⋊8M4(2)
 Chief series C1 — C5 — C52 — C5×C10 — C52⋊6C4 — C52⋊5C8 — C52⋊8M4(2)
 Lower central C52 — C5×C10 — C52⋊8M4(2)
 Upper central C1 — C2 — C4

Generators and relations for C528M4(2)
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a3, dad=a-1, cbc-1=b2, dbd=b-1, dcd=c5 >

Subgroups: 412 in 56 conjugacy classes, 18 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C8, C2×C4, D5, C10, C10, M4(2), Dic5, C20, C20, D10, C52, C5⋊C8, C4×D5, C5⋊D5, C5×C10, C4.F5, C526C4, C5×C20, C2×C5⋊D5, C525C8, C4×C5⋊D5, C528M4(2)
Quotients: C1, C2, C4, C22, C2×C4, M4(2), F5, C2×F5, C4.F5, C52⋊C4, C2×C52⋊C4, C528M4(2)

Smallest permutation representation of C528M4(2)
On 40 points
Generators in S40
```(1 37 10 26 17)(2 27 38 18 11)(3 19 28 12 39)(4 13 20 40 29)(5 33 14 30 21)(6 31 34 22 15)(7 23 32 16 35)(8 9 24 36 25)
(1 10 17 37 26)(2 18 27 11 38)(3 28 39 19 12)(4 40 13 29 20)(5 14 21 33 30)(6 22 31 15 34)(7 32 35 23 16)(8 36 9 25 24)
(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 29)(10 26)(11 31)(12 28)(13 25)(14 30)(15 27)(16 32)(17 37)(18 34)(19 39)(20 36)(21 33)(22 38)(23 35)(24 40)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A ··· 5F 8A 8B 8C 8D 10A ··· 10F 20A ··· 20L order 1 2 2 4 4 4 5 ··· 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 50 2 25 25 4 ··· 4 50 50 50 50 4 ··· 4 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 4 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 M4(2) F5 C2×F5 C4.F5 C52⋊C4 C2×C52⋊C4 C52⋊8M4(2) kernel C52⋊8M4(2) C52⋊5C8 C4×C5⋊D5 C5×C20 C2×C5⋊D5 C52 C20 C10 C5 C4 C2 C1 # reps 1 2 1 2 2 2 2 2 4 4 4 8

Matrix representation of C528M4(2) in GL4(𝔽41) generated by

 0 7 0 0 35 6 0 0 1 7 34 40 35 6 1 0
,
 40 1 0 0 5 35 0 0 6 1 40 34 39 35 7 7
,
 7 1 33 40 40 35 6 7 26 33 40 0 17 33 40 0
,
 6 1 0 0 6 35 0 0 38 23 40 0 31 22 7 1
`G:=sub<GL(4,GF(41))| [0,35,1,35,7,6,7,6,0,0,34,1,0,0,40,0],[40,5,6,39,1,35,1,35,0,0,40,7,0,0,34,7],[7,40,26,17,1,35,33,33,33,6,40,40,40,7,0,0],[6,6,38,31,1,35,23,22,0,0,40,7,0,0,0,1] >;`

C528M4(2) in GAP, Magma, Sage, TeX

`C_5^2\rtimes_8M_4(2)`
`% in TeX`

`G:=Group("C5^2:8M4(2)");`
`// GroupNames label`

`G:=SmallGroup(400,157);`
`// by ID`

`G=gap.SmallGroup(400,157);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,50,1444,496,5765,2897]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^3,d*a*d=a^-1,c*b*c^-1=b^2,d*b*d=b^-1,d*c*d=c^5>;`
`// generators/relations`

׿
×
𝔽