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## G = 2- 1+4.C10order 320 = 26·5

### The non-split extension by 2- 1+4 of C10 acting via C10/C2=C5

Aliases: 2- 1+4.C10, C4.(C24⋊C5), C2.C25⋊C5, 2- 1+4⋊C52C2, C2.3(C2×C24⋊C5), SmallGroup(320,1586)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2- 1+4 — 2- 1+4.C10
 Chief series C1 — C2 — 2- 1+4 — 2- 1+4⋊C5 — 2- 1+4.C10
 Lower central 2- 1+4 — 2- 1+4.C10
 Upper central C1 — C4

Generators and relations for 2- 1+4.C10
G = < a,b,c,d,e | a4=b2=1, c2=d2=e10=a2, bab=a-1, ebe-1=ac=ca, ad=da, eae-1=cd, bc=cb, bd=db, dcd-1=a2c, ece-1=a-1bcd, ede-1=a >

Subgroups: 515 in 90 conjugacy classes, 7 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C2×C4○D4, 2+ 1+4, 2- 1+4, 2- 1+4, C2.C25, 2- 1+4⋊C5, 2- 1+4.C10
Quotients: C1, C2, C5, C10, C24⋊C5, C2×C24⋊C5, 2- 1+4.C10

Character table of 2- 1+4.C10

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 10A 10B 10C 10D 20A 20B 20C 20D 20E 20F 20G 20H size 1 1 10 10 10 1 1 10 10 10 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 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 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 ζ52 ζ54 ζ53 ζ5 ζ52 ζ54 ζ53 ζ5 ζ54 ζ5 ζ52 ζ52 ζ53 ζ53 ζ54 ζ5 linear of order 5 ρ4 1 1 1 1 1 1 1 1 1 1 ζ54 ζ53 ζ5 ζ52 ζ54 ζ53 ζ5 ζ52 ζ53 ζ52 ζ54 ζ54 ζ5 ζ5 ζ53 ζ52 linear of order 5 ρ5 1 1 1 1 1 1 1 1 1 1 ζ5 ζ52 ζ54 ζ53 ζ5 ζ52 ζ54 ζ53 ζ52 ζ53 ζ5 ζ5 ζ54 ζ54 ζ52 ζ53 linear of order 5 ρ6 1 1 1 -1 -1 -1 -1 -1 1 1 ζ53 ζ5 ζ52 ζ54 ζ53 ζ5 ζ52 ζ54 -ζ5 -ζ54 -ζ53 -ζ53 -ζ52 -ζ52 -ζ5 -ζ54 linear of order 10 ρ7 1 1 1 1 1 1 1 1 1 1 ζ53 ζ5 ζ52 ζ54 ζ53 ζ5 ζ52 ζ54 ζ5 ζ54 ζ53 ζ53 ζ52 ζ52 ζ5 ζ54 linear of order 5 ρ8 1 1 1 -1 -1 -1 -1 -1 1 1 ζ54 ζ53 ζ5 ζ52 ζ54 ζ53 ζ5 ζ52 -ζ53 -ζ52 -ζ54 -ζ54 -ζ5 -ζ5 -ζ53 -ζ52 linear of order 10 ρ9 1 1 1 -1 -1 -1 -1 -1 1 1 ζ52 ζ54 ζ53 ζ5 ζ52 ζ54 ζ53 ζ5 -ζ54 -ζ5 -ζ52 -ζ52 -ζ53 -ζ53 -ζ54 -ζ5 linear of order 10 ρ10 1 1 1 -1 -1 -1 -1 -1 1 1 ζ5 ζ52 ζ54 ζ53 ζ5 ζ52 ζ54 ζ53 -ζ52 -ζ53 -ζ5 -ζ5 -ζ54 -ζ54 -ζ52 -ζ53 linear of order 10 ρ11 4 -4 0 0 0 4i -4i 0 0 0 -1 -1 -1 -1 1 1 1 1 i i -i i -i i -i -i complex faithful ρ12 4 -4 0 0 0 -4i 4i 0 0 0 -1 -1 -1 -1 1 1 1 1 -i -i i -i i -i i i complex faithful ρ13 4 -4 0 0 0 4i -4i 0 0 0 -ζ52 -ζ54 -ζ53 -ζ5 ζ52 ζ54 ζ53 ζ5 ζ4ζ54 ζ4ζ5 ζ43ζ52 ζ4ζ52 ζ43ζ53 ζ4ζ53 ζ43ζ54 ζ43ζ5 complex faithful ρ14 4 -4 0 0 0 4i -4i 0 0 0 -ζ54 -ζ53 -ζ5 -ζ52 ζ54 ζ53 ζ5 ζ52 ζ4ζ53 ζ4ζ52 ζ43ζ54 ζ4ζ54 ζ43ζ5 ζ4ζ5 ζ43ζ53 ζ43ζ52 complex faithful ρ15 4 -4 0 0 0 -4i 4i 0 0 0 -ζ53 -ζ5 -ζ52 -ζ54 ζ53 ζ5 ζ52 ζ54 ζ43ζ5 ζ43ζ54 ζ4ζ53 ζ43ζ53 ζ4ζ52 ζ43ζ52 ζ4ζ5 ζ4ζ54 complex faithful ρ16 4 -4 0 0 0 -4i 4i 0 0 0 -ζ54 -ζ53 -ζ5 -ζ52 ζ54 ζ53 ζ5 ζ52 ζ43ζ53 ζ43ζ52 ζ4ζ54 ζ43ζ54 ζ4ζ5 ζ43ζ5 ζ4ζ53 ζ4ζ52 complex faithful ρ17 4 -4 0 0 0 -4i 4i 0 0 0 -ζ5 -ζ52 -ζ54 -ζ53 ζ5 ζ52 ζ54 ζ53 ζ43ζ52 ζ43ζ53 ζ4ζ5 ζ43ζ5 ζ4ζ54 ζ43ζ54 ζ4ζ52 ζ4ζ53 complex faithful ρ18 4 -4 0 0 0 4i -4i 0 0 0 -ζ5 -ζ52 -ζ54 -ζ53 ζ5 ζ52 ζ54 ζ53 ζ4ζ52 ζ4ζ53 ζ43ζ5 ζ4ζ5 ζ43ζ54 ζ4ζ54 ζ43ζ52 ζ43ζ53 complex faithful ρ19 4 -4 0 0 0 4i -4i 0 0 0 -ζ53 -ζ5 -ζ52 -ζ54 ζ53 ζ5 ζ52 ζ54 ζ4ζ5 ζ4ζ54 ζ43ζ53 ζ4ζ53 ζ43ζ52 ζ4ζ52 ζ43ζ5 ζ43ζ54 complex faithful ρ20 4 -4 0 0 0 -4i 4i 0 0 0 -ζ52 -ζ54 -ζ53 -ζ5 ζ52 ζ54 ζ53 ζ5 ζ43ζ54 ζ43ζ5 ζ4ζ52 ζ43ζ52 ζ4ζ53 ζ43ζ53 ζ4ζ54 ζ4ζ5 complex faithful ρ21 5 5 1 -1 3 -5 -5 -1 -3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C24⋊C5 ρ22 5 5 1 -3 1 5 5 1 1 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ23 5 5 -3 1 1 5 5 -3 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ24 5 5 1 3 -1 -5 -5 -1 1 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C24⋊C5 ρ25 5 5 -3 -1 -1 -5 -5 3 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C24⋊C5 ρ26 5 5 1 1 -3 5 5 1 -3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5

Smallest permutation representation of 2- 1+4.C10
On 64 points
Generators in S64
```(1 34 3 44)(2 39 4 29)(5 23 15 13)(6 28 16 38)(7 25 17 35)(8 20 18 10)(9 48 19 58)(11 33 21 43)(12 30 22 40)(14 53 24 63)(26 59 36 49)(27 62 37 52)(31 64 41 54)(32 47 42 57)(45 51 55 61)(46 50 56 60)
(1 10)(2 15)(3 20)(4 5)(6 57)(7 41)(8 44)(9 46)(11 62)(12 26)(13 29)(14 51)(16 47)(17 31)(18 34)(19 56)(21 52)(22 36)(23 39)(24 61)(25 64)(27 33)(28 42)(30 49)(32 38)(35 54)(37 43)(40 59)(45 53)(48 60)(50 58)(55 63)
(1 63 3 53)(2 48 4 58)(5 50 15 60)(6 22 16 12)(7 21 17 11)(8 61 18 51)(9 39 19 29)(10 55 20 45)(13 46 23 56)(14 44 24 34)(25 43 35 33)(26 57 36 47)(27 64 37 54)(28 40 38 30)(31 62 41 52)(32 49 42 59)
(1 30 3 40)(2 35 4 25)(5 64 15 54)(6 24 16 14)(7 29 17 39)(8 26 18 36)(9 21 19 11)(10 49 20 59)(12 34 22 44)(13 31 23 41)(27 60 37 50)(28 63 38 53)(32 45 42 55)(33 48 43 58)(46 52 56 62)(47 51 57 61)
(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 41 42 43 44)(45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)```

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

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

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

Matrix representation of 2- 1+4.C10 in GL4(𝔽5) generated by

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

2- 1+4.C10 in GAP, Magma, Sage, TeX

`2_-^{1+4}.C_{10}`
`% in TeX`

`G:=Group("ES-(2,2).C10");`
`// GroupNames label`

`G:=SmallGroup(320,1586);`
`// by ID`

`G=gap.SmallGroup(320,1586);`
`# by ID`

`G:=PCGroup([7,-2,-5,-2,2,2,2,-2,1120,849,1270,521,248,1936,718,375,172,3162,1027]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^4=b^2=1,c^2=d^2=e^10=a^2,b*a*b=a^-1,e*b*e^-1=a*c=c*a,a*d=d*a,e*a*e^-1=c*d,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=a^-1*b*c*d,e*d*e^-1=a>;`
`// generators/relations`

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