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G = 2- 1+4⋊C5order 160 = 25·5

The semidirect product of 2- 1+4 and C5 acting faithfully

Aliases: 2- 1+4⋊C5, C2.(C24⋊C5), SmallGroup(160,199)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2- 1+4 — 2- 1+4⋊C5
 Chief series C1 — C2 — 2- 1+4 — 2- 1+4⋊C5
 Lower central 2- 1+4 — 2- 1+4⋊C5
 Upper central C1 — C2

Generators and relations for 2- 1+4⋊C5
G = < a,b,c,d,e | a4=b2=e5=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, eae-1=a2bcd, bc=cb, bd=db, ebe-1=ab, dcd-1=a2c, ece-1=abc, ede-1=cd >

10C2
16C5
5C4
5C4
5C22
16C10
5Q8
5D4
5Q8
5D4

Character table of 2- 1+4⋊C5

 class 1 2A 2B 4A 4B 5A 5B 5C 5D 10A 10B 10C 10D size 1 1 10 10 10 16 16 16 16 16 16 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 ζ53 ζ5 ζ54 ζ52 ζ52 ζ54 ζ5 ζ53 linear of order 5 ρ3 1 1 1 1 1 ζ54 ζ53 ζ52 ζ5 ζ5 ζ52 ζ53 ζ54 linear of order 5 ρ4 1 1 1 1 1 ζ5 ζ52 ζ53 ζ54 ζ54 ζ53 ζ52 ζ5 linear of order 5 ρ5 1 1 1 1 1 ζ52 ζ54 ζ5 ζ53 ζ53 ζ5 ζ54 ζ52 linear of order 5 ρ6 4 -4 0 0 0 -1 -1 -1 -1 1 1 1 1 symplectic faithful, Schur index 2 ρ7 4 -4 0 0 0 -ζ54 -ζ53 -ζ52 -ζ5 ζ5 ζ52 ζ53 ζ54 complex faithful ρ8 4 -4 0 0 0 -ζ52 -ζ54 -ζ5 -ζ53 ζ53 ζ5 ζ54 ζ52 complex faithful ρ9 4 -4 0 0 0 -ζ53 -ζ5 -ζ54 -ζ52 ζ52 ζ54 ζ5 ζ53 complex faithful ρ10 4 -4 0 0 0 -ζ5 -ζ52 -ζ53 -ζ54 ζ54 ζ53 ζ52 ζ5 complex faithful ρ11 5 5 1 1 -3 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ12 5 5 1 -3 1 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ13 5 5 -3 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5

Smallest permutation representation of 2- 1+4⋊C5
On 32 points
Generators in S32
```(1 11 2 30)(3 10 17 29)(4 31 13 12)(5 7 14 16)(6 27 15 21)(8 23 32 22)(9 26 28 20)(18 19 24 25)
(1 25)(2 19)(3 22)(4 28)(5 27)(6 7)(8 29)(9 13)(10 32)(11 24)(12 20)(14 21)(15 16)(17 23)(18 30)(26 31)
(1 6 2 15)(3 12 17 31)(4 29 13 10)(5 18 14 24)(7 19 16 25)(8 9 32 28)(11 27 30 21)(20 23 26 22)
(1 9 2 28)(3 5 17 14)(4 25 13 19)(6 8 15 32)(7 29 16 10)(11 26 30 20)(12 24 31 18)(21 22 27 23)
(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)```

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

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

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

2- 1+4⋊C5 is a maximal subgroup of   2- 1+4.D5  2- 1+4⋊D5  2- 1+4.C10
2- 1+4⋊C5 is a maximal quotient of   C22.58C24⋊C5

Matrix representation of 2- 1+4⋊C5 in GL4(𝔽3) generated by

 2 0 1 1 2 0 1 2 0 1 1 1 1 2 0 0
,
 0 0 0 1 2 0 1 2 1 1 0 1 1 0 0 0
,
 2 1 0 2 2 1 1 0 0 2 2 2 1 0 1 1
,
 2 1 0 1 1 1 2 0 0 0 2 1 0 0 1 1
,
 0 1 2 1 1 2 0 1 0 1 2 0 0 0 2 1
`G:=sub<GL(4,GF(3))| [2,2,0,1,0,0,1,2,1,1,1,0,1,2,1,0],[0,2,1,1,0,0,1,0,0,1,0,0,1,2,1,0],[2,2,0,1,1,1,2,0,0,1,2,1,2,0,2,1],[2,1,0,0,1,1,0,0,0,2,2,1,1,0,1,1],[0,1,0,0,1,2,1,0,2,0,2,2,1,1,0,1] >;`

2- 1+4⋊C5 in GAP, Magma, Sage, TeX

`2_-^{1+4}\rtimes C_5`
`% in TeX`

`G:=Group("ES-(2,2):C5");`
`// GroupNames label`

`G:=SmallGroup(160,199);`
`// by ID`

`G=gap.SmallGroup(160,199);`
`# by ID`

`G:=PCGroup([6,-5,-2,2,2,2,-2,481,812,332,158,1323,489,255,117,2254,730]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^4=b^2=e^5=1,c^2=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^2*b*c*d,b*c=c*b,b*d=d*b,e*b*e^-1=a*b,d*c*d^-1=a^2*c,e*c*e^-1=a*b*c,e*d*e^-1=c*d>;`
`// generators/relations`

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