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## G = 2- 1+4.D5order 320 = 26·5

### The non-split extension by 2- 1+4 of D5 acting faithfully

Aliases: 2- 1+4.D5, C2.2(C24⋊D5), 2- 1+4⋊C5.C2, SmallGroup(320,1581)

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=1, c2=d2=f2=a2, bab=a-1, ac=ca, ebe-1=ad=da, eae-1=fdf-1=a-1bd, faf-1=ede-1=bcd, bc=cb, bd=db, fbf-1=acd, dcd-1=a2c, ece-1=a2bc, fcf-1=abc, fef-1=e-1 >

10C2
16C5
5C22
5C4
5C4
10C4
10C4
20C4
16C10
5Q8
5D4
5Q8
5D4
10C2×C4
10C2×C4
10Q8
10C8
10C8
10Q8
16Dic5
5C42
10Q16
10SD16
10Q16
10C4⋊C4
10C4⋊C4
10SD16

Character table of 2- 1+4.D5

 class 1 2A 2B 4A 4B 4C 4D 4E 5A 5B 8A 8B 10A 10B size 1 1 10 10 10 20 20 40 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 2 2 0 0 0 -1-√5/2 -1+√5/2 0 0 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ4 2 2 2 2 2 0 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 2 -2 0 -1 -1 0 0 1 1 symplectic faithful, Schur index 2 ρ6 4 -4 0 0 0 -2 2 0 -1 -1 0 0 1 1 symplectic faithful, Schur index 2 ρ7 5 5 1 1 -3 1 1 -1 0 0 -1 1 0 0 orthogonal lifted from C24⋊D5 ρ8 5 5 -3 1 1 1 1 1 0 0 -1 -1 0 0 orthogonal lifted from C24⋊D5 ρ9 5 5 1 -3 1 -1 -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 -3 1 1 -1 -1 -1 0 0 1 1 0 0 orthogonal lifted from C24⋊D5 ρ12 5 5 1 -3 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 symplectic faithful, Schur index 2 ρ14 8 -8 0 0 0 0 0 0 1+√5/2 1-√5/2 0 0 -1-√5/2 -1+√5/2 symplectic faithful, Schur index 2

Smallest permutation representation of 2- 1+4.D5
On 64 points
Generators in S64
```(1 59 3 45)(2 10 4 20)(5 32 30 7)(6 15 31 26)(8 14 33 24)(9 21 34 11)(12 27 22 16)(13 25 23 19)(17 29 28 18)(35 46 50 55)(36 53 51 38)(37 40 52 63)(39 58 54 49)(41 56 64 47)(42 61 60 43)(44 57 62 48)
(1 42)(2 17)(3 60)(4 28)(5 31)(6 30)(7 26)(8 22)(9 19)(10 18)(11 13)(12 33)(14 27)(15 32)(16 24)(20 29)(21 23)(25 34)(35 44)(36 52)(37 51)(38 63)(39 47)(40 53)(41 49)(43 59)(45 61)(46 48)(50 62)(54 56)(55 57)(58 64)
(1 52 3 37)(2 31 4 6)(5 28 30 17)(7 29 32 18)(8 11 33 21)(9 24 34 14)(10 26 20 15)(12 23 22 13)(16 25 27 19)(35 49 50 58)(36 60 51 42)(38 61 53 43)(39 55 54 46)(40 59 63 45)(41 62 64 44)(47 57 56 48)
(1 47 3 56)(2 22 4 12)(5 21 30 11)(6 13 31 23)(7 9 32 34)(8 28 33 17)(10 16 20 27)(14 18 24 29)(15 25 26 19)(35 53 50 38)(36 46 51 55)(37 57 52 48)(39 60 54 42)(40 62 63 44)(41 45 64 59)(43 49 61 58)
(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)
(1 4 3 2)(5 54 30 39)(6 53 31 38)(7 52 32 37)(8 51 33 36)(9 50 34 35)(10 46 20 55)(11 45 21 59)(12 49 22 58)(13 48 23 57)(14 47 24 56)(15 41 26 64)(16 40 27 63)(17 44 28 62)(18 43 29 61)(19 42 25 60)```

`G:=sub<Sym(64)| (1,59,3,45)(2,10,4,20)(5,32,30,7)(6,15,31,26)(8,14,33,24)(9,21,34,11)(12,27,22,16)(13,25,23,19)(17,29,28,18)(35,46,50,55)(36,53,51,38)(37,40,52,63)(39,58,54,49)(41,56,64,47)(42,61,60,43)(44,57,62,48), (1,42)(2,17)(3,60)(4,28)(5,31)(6,30)(7,26)(8,22)(9,19)(10,18)(11,13)(12,33)(14,27)(15,32)(16,24)(20,29)(21,23)(25,34)(35,44)(36,52)(37,51)(38,63)(39,47)(40,53)(41,49)(43,59)(45,61)(46,48)(50,62)(54,56)(55,57)(58,64), (1,52,3,37)(2,31,4,6)(5,28,30,17)(7,29,32,18)(8,11,33,21)(9,24,34,14)(10,26,20,15)(12,23,22,13)(16,25,27,19)(35,49,50,58)(36,60,51,42)(38,61,53,43)(39,55,54,46)(40,59,63,45)(41,62,64,44)(47,57,56,48), (1,47,3,56)(2,22,4,12)(5,21,30,11)(6,13,31,23)(7,9,32,34)(8,28,33,17)(10,16,20,27)(14,18,24,29)(15,25,26,19)(35,53,50,38)(36,46,51,55)(37,57,52,48)(39,60,54,42)(40,62,63,44)(41,45,64,59)(43,49,61,58), (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), (1,4,3,2)(5,54,30,39)(6,53,31,38)(7,52,32,37)(8,51,33,36)(9,50,34,35)(10,46,20,55)(11,45,21,59)(12,49,22,58)(13,48,23,57)(14,47,24,56)(15,41,26,64)(16,40,27,63)(17,44,28,62)(18,43,29,61)(19,42,25,60)>;`

`G:=Group( (1,59,3,45)(2,10,4,20)(5,32,30,7)(6,15,31,26)(8,14,33,24)(9,21,34,11)(12,27,22,16)(13,25,23,19)(17,29,28,18)(35,46,50,55)(36,53,51,38)(37,40,52,63)(39,58,54,49)(41,56,64,47)(42,61,60,43)(44,57,62,48), (1,42)(2,17)(3,60)(4,28)(5,31)(6,30)(7,26)(8,22)(9,19)(10,18)(11,13)(12,33)(14,27)(15,32)(16,24)(20,29)(21,23)(25,34)(35,44)(36,52)(37,51)(38,63)(39,47)(40,53)(41,49)(43,59)(45,61)(46,48)(50,62)(54,56)(55,57)(58,64), (1,52,3,37)(2,31,4,6)(5,28,30,17)(7,29,32,18)(8,11,33,21)(9,24,34,14)(10,26,20,15)(12,23,22,13)(16,25,27,19)(35,49,50,58)(36,60,51,42)(38,61,53,43)(39,55,54,46)(40,59,63,45)(41,62,64,44)(47,57,56,48), (1,47,3,56)(2,22,4,12)(5,21,30,11)(6,13,31,23)(7,9,32,34)(8,28,33,17)(10,16,20,27)(14,18,24,29)(15,25,26,19)(35,53,50,38)(36,46,51,55)(37,57,52,48)(39,60,54,42)(40,62,63,44)(41,45,64,59)(43,49,61,58), (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), (1,4,3,2)(5,54,30,39)(6,53,31,38)(7,52,32,37)(8,51,33,36)(9,50,34,35)(10,46,20,55)(11,45,21,59)(12,49,22,58)(13,48,23,57)(14,47,24,56)(15,41,26,64)(16,40,27,63)(17,44,28,62)(18,43,29,61)(19,42,25,60) );`

`G=PermutationGroup([[(1,59,3,45),(2,10,4,20),(5,32,30,7),(6,15,31,26),(8,14,33,24),(9,21,34,11),(12,27,22,16),(13,25,23,19),(17,29,28,18),(35,46,50,55),(36,53,51,38),(37,40,52,63),(39,58,54,49),(41,56,64,47),(42,61,60,43),(44,57,62,48)], [(1,42),(2,17),(3,60),(4,28),(5,31),(6,30),(7,26),(8,22),(9,19),(10,18),(11,13),(12,33),(14,27),(15,32),(16,24),(20,29),(21,23),(25,34),(35,44),(36,52),(37,51),(38,63),(39,47),(40,53),(41,49),(43,59),(45,61),(46,48),(50,62),(54,56),(55,57),(58,64)], [(1,52,3,37),(2,31,4,6),(5,28,30,17),(7,29,32,18),(8,11,33,21),(9,24,34,14),(10,26,20,15),(12,23,22,13),(16,25,27,19),(35,49,50,58),(36,60,51,42),(38,61,53,43),(39,55,54,46),(40,59,63,45),(41,62,64,44),(47,57,56,48)], [(1,47,3,56),(2,22,4,12),(5,21,30,11),(6,13,31,23),(7,9,32,34),(8,28,33,17),(10,16,20,27),(14,18,24,29),(15,25,26,19),(35,53,50,38),(36,46,51,55),(37,57,52,48),(39,60,54,42),(40,62,63,44),(41,45,64,59),(43,49,61,58)], [(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)], [(1,4,3,2),(5,54,30,39),(6,53,31,38),(7,52,32,37),(8,51,33,36),(9,50,34,35),(10,46,20,55),(11,45,21,59),(12,49,22,58),(13,48,23,57),(14,47,24,56),(15,41,26,64),(16,40,27,63),(17,44,28,62),(18,43,29,61),(19,42,25,60)]])`

Matrix representation of 2- 1+4.D5 in GL4(𝔽3) generated by

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

2- 1+4.D5 in GAP, Magma, Sage, TeX

`2_-^{1+4}.D_5`
`% in TeX`

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

`G:=SmallGroup(320,1581);`
`// by ID`

`G=gap.SmallGroup(320,1581);`
`# by ID`

`G:=PCGroup([7,-2,-5,-2,2,2,2,-2,1120,113,632,324,5043,850,521,248,3854,2111,718,375,172,2105,3582,1027]);`
`// Polycyclic`

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

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