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G = 2- 1+4⋊C5order 160 = 25·5

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

non-abelian, soluble

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

Series: Derived Chief Lower central Upper central

C1C22- 1+4 — 2- 1+4⋊C5
C1C22- 1+4 — 2- 1+4⋊C5
2- 1+4 — 2- 1+4⋊C5
C1C2

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
5C2×C4
5C2×C4
5D4
5C2×C4
5Q8
5D4
5C4○D4
5C4○D4
5C2×Q8

Character table of 2- 1+4⋊C5

 class 12A2B4A4B5A5B5C5D10A10B10C10D
 size 111010101616161616161616
ρ11111111111111    trivial
ρ211111ζ53ζ5ζ54ζ52ζ52ζ54ζ5ζ53    linear of order 5
ρ311111ζ54ζ53ζ52ζ5ζ5ζ52ζ53ζ54    linear of order 5
ρ411111ζ5ζ52ζ53ζ54ζ54ζ53ζ52ζ5    linear of order 5
ρ511111ζ52ζ54ζ5ζ53ζ53ζ5ζ54ζ52    linear of order 5
ρ64-4000-1-1-1-11111    symplectic faithful, Schur index 2
ρ74-40005453525ζ5ζ52ζ53ζ54    complex faithful
ρ84-40005254553ζ53ζ5ζ54ζ52    complex faithful
ρ94-40005355452ζ52ζ54ζ5ζ53    complex faithful
ρ104-40005525354ζ54ζ53ζ52ζ5    complex faithful
ρ115511-300000000    orthogonal lifted from C24⋊C5
ρ12551-3100000000    orthogonal lifted from C24⋊C5
ρ1355-31100000000    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

2011
2012
0111
1200
,
0001
2012
1101
1000
,
2102
2110
0222
1011
,
2101
1120
0021
0011
,
0121
1201
0120
0021
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

Export

Subgroup lattice of 2- 1+4⋊C5 in TeX
Character table of 2- 1+4⋊C5 in TeX

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