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## G = 2- 1+4⋊D5order 320 = 26·5

### The semidirect product of 2- 1+4 and D5 acting faithfully

Aliases: 2- 1+4⋊D5, 2- 1+4⋊C5⋊C2, C2.3(C24⋊D5), SmallGroup(320,1582)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2- 1+4 — 2- 1+4⋊C5 — 2- 1+4⋊D5
 Chief series C1 — C2 — 2- 1+4 — 2- 1+4⋊C5 — 2- 1+4⋊D5
 Lower central 2- 1+4⋊C5 — 2- 1+4⋊D5
 Upper central C1 — C2

Generators and relations for 2- 1+4⋊D5
G = < a,b,c,d,e,f | a4=b2=e5=f2=1, c2=d2=a2, bab=a-1, ac=ca, ebe-1=ad=da, eae-1=fdf=a-1bd, faf=ede-1=bcd, bc=cb, bd=db, fbf=acd, dcd-1=a2c, ece-1=a2bc, fcf=abc, fef=e-1 >

10C2
40C2
16C5
5C4
5C4
5C22
20C22
20C22
20C4
20C22
16C10
16D5
16D5
5D4
5Q8
5Q8
5D4
10C23
10D4
10D4
10C8
10C2×C4
10C8
16D10
5C42
10D8
10C22⋊C4
10C22⋊C4
10SD16
10D8
10SD16

Character table of 2- 1+4⋊D5

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 8A 8B 10A 10B size 1 1 10 40 10 10 20 20 32 32 40 40 32 32 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ3 2 2 2 0 2 2 0 0 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ4 2 2 2 0 2 2 0 0 -1+√5/2 -1-√5/2 0 0 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ5 4 -4 0 0 0 0 2i -2i -1 -1 0 0 1 1 complex faithful ρ6 4 -4 0 0 0 0 -2i 2i -1 -1 0 0 1 1 complex faithful ρ7 5 5 -3 -1 1 1 -1 -1 0 0 1 1 0 0 orthogonal lifted from C24⋊D5 ρ8 5 5 1 -1 -3 1 1 1 0 0 -1 1 0 0 orthogonal lifted from C24⋊D5 ρ9 5 5 1 -1 1 -3 1 1 0 0 1 -1 0 0 orthogonal lifted from C24⋊D5 ρ10 5 5 1 1 -3 1 -1 -1 0 0 1 -1 0 0 orthogonal lifted from C24⋊D5 ρ11 5 5 1 1 1 -3 -1 -1 0 0 -1 1 0 0 orthogonal lifted from C24⋊D5 ρ12 5 5 -3 1 1 1 1 1 0 0 -1 -1 0 0 orthogonal lifted from C24⋊D5 ρ13 8 -8 0 0 0 0 0 0 1-√5/2 1+√5/2 0 0 -1+√5/2 -1-√5/2 orthogonal faithful ρ14 8 -8 0 0 0 0 0 0 1+√5/2 1-√5/2 0 0 -1-√5/2 -1+√5/2 orthogonal faithful

Smallest permutation representation of 2- 1+4⋊D5
On 32 points
Generators in S32
(1 18 2 11)(3 20 32 8)(4 29 28 5)(6 21 30 9)(7 14 31 27)(10 24 22 16)(12 17 19 25)(13 23 26 15)
(1 4)(2 28)(3 10)(5 18)(6 25)(7 15)(8 24)(9 12)(11 29)(13 27)(14 26)(16 20)(17 30)(19 21)(22 32)(23 31)
(1 14 2 27)(3 30 32 6)(4 26 28 13)(5 23 29 15)(7 18 31 11)(8 21 20 9)(10 17 22 25)(12 24 19 16)
(1 8 2 20)(3 11 32 18)(4 24 28 16)(5 10 29 22)(6 7 30 31)(9 27 21 14)(12 13 19 26)(15 17 23 25)
(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)
(1 2)(3 31)(4 30)(5 29)(6 28)(7 32)(8 10)(11 12)(13 16)(14 15)(18 19)(20 22)(23 27)(24 26)

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

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

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

Matrix representation of 2- 1+4⋊D5 in GL4(𝔽5) generated by

 1 0 0 2 0 0 4 3 2 1 0 2 4 0 0 4
,
 3 2 3 4 2 1 4 0 3 4 0 3 1 0 2 1
,
 2 0 3 0 1 0 2 1 0 0 3 0 3 4 1 0
,
 2 1 3 1 2 3 0 3 1 3 2 4 0 1 3 3
,
 1 2 2 2 0 0 1 2 0 1 0 3 0 1 3 3
,
 1 3 3 0 0 4 0 0 0 0 4 0 0 2 4 1
G:=sub<GL(4,GF(5))| [1,0,2,4,0,0,1,0,0,4,0,0,2,3,2,4],[3,2,3,1,2,1,4,0,3,4,0,2,4,0,3,1],[2,1,0,3,0,0,0,4,3,2,3,1,0,1,0,0],[2,2,1,0,1,3,3,1,3,0,2,3,1,3,4,3],[1,0,0,0,2,0,1,1,2,1,0,3,2,2,3,3],[1,0,0,0,3,4,0,2,3,0,4,4,0,0,0,1] >;

2- 1+4⋊D5 in GAP, Magma, Sage, TeX

2_-^{1+4}\rtimes D_5
% in TeX

G:=Group("ES-(2,2):D5");
// GroupNames label

G:=SmallGroup(320,1582);
// by ID

G=gap.SmallGroup(320,1582);
# by ID

G:=PCGroup([7,-2,-5,-2,2,2,2,-2,113,632,324,5043,850,521,248,3854,2111,718,375,172,2105,3582,1027]);
// Polycyclic

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

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