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## G = 2- 1+4.2D5order 320 = 26·5

### The non-split extension by 2- 1+4 of D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — 2- 1+4.2D5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×Dic10 — D4.9D10 — 2- 1+4.2D5
 Lower central C5 — C10 — C2×C20 — 2- 1+4.2D5
 Upper central C1 — C2 — C2×C4 — 2- 1+4

Generators and relations for 2- 1+4.2D5
G = < a,b,c,d,e,f | a4=b2=e5=1, c2=d2=f2=a2, bab=faf-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf-1=ab, dcd-1=fcf-1=a2c, ce=ec, de=ed, fdf-1=a2cd, fef-1=e-1 >

Subgroups: 414 in 142 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C52C8, Dic10, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, D4.10D4, C4.Dic5, C4×Dic5, C10.D4, D4.D5, C5⋊Q16, C2×Dic10, Q8×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C20.10D4, D42Dic5, Dic5⋊Q8, D4.9D10, C5×2- 1+4, 2- 1+4.2D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, D4.10D4, C2×C5⋊D4, C242D5, 2- 1+4.2D5

Smallest permutation representation of 2- 1+4.2D5
On 80 points
Generators in S80
```(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)
(1 59)(2 60)(3 56)(4 57)(5 58)(6 51)(7 52)(8 53)(9 54)(10 55)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 79 9 74)(2 80 10 75)(3 76 6 71)(4 77 7 72)(5 78 8 73)(11 61 16 66)(12 62 17 67)(13 63 18 68)(14 64 19 69)(15 65 20 70)(21 56 26 51)(22 57 27 52)(23 58 28 53)(24 59 29 54)(25 60 30 55)(31 41 36 46)(32 42 37 47)(33 43 38 48)(34 44 39 49)(35 45 40 50)
(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 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)
(1 23 9 28)(2 22 10 27)(3 21 6 26)(4 25 7 30)(5 24 8 29)(11 31 16 36)(12 35 17 40)(13 34 18 39)(14 33 19 38)(15 32 20 37)(41 76 46 71)(42 80 47 75)(43 79 48 74)(44 78 49 73)(45 77 50 72)(51 61 56 66)(52 65 57 70)(53 64 58 69)(54 63 59 68)(55 62 60 67)```

`G:=sub<Sym(80)| (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (1,59)(2,60)(3,56)(4,57)(5,58)(6,51)(7,52)(8,53)(9,54)(10,55)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,79,9,74)(2,80,10,75)(3,76,6,71)(4,77,7,72)(5,78,8,73)(11,61,16,66)(12,62,17,67)(13,63,18,68)(14,64,19,69)(15,65,20,70)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55)(31,41,36,46)(32,42,37,47)(33,43,38,48)(34,44,39,49)(35,45,40,50), (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)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,23,9,28)(2,22,10,27)(3,21,6,26)(4,25,7,30)(5,24,8,29)(11,31,16,36)(12,35,17,40)(13,34,18,39)(14,33,19,38)(15,32,20,37)(41,76,46,71)(42,80,47,75)(43,79,48,74)(44,78,49,73)(45,77,50,72)(51,61,56,66)(52,65,57,70)(53,64,58,69)(54,63,59,68)(55,62,60,67)>;`

`G:=Group( (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (1,59)(2,60)(3,56)(4,57)(5,58)(6,51)(7,52)(8,53)(9,54)(10,55)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65), (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,79,9,74)(2,80,10,75)(3,76,6,71)(4,77,7,72)(5,78,8,73)(11,61,16,66)(12,62,17,67)(13,63,18,68)(14,64,19,69)(15,65,20,70)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55)(31,41,36,46)(32,42,37,47)(33,43,38,48)(34,44,39,49)(35,45,40,50), (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)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,23,9,28)(2,22,10,27)(3,21,6,26)(4,25,7,30)(5,24,8,29)(11,31,16,36)(12,35,17,40)(13,34,18,39)(14,33,19,38)(15,32,20,37)(41,76,46,71)(42,80,47,75)(43,79,48,74)(44,78,49,73)(45,77,50,72)(51,61,56,66)(52,65,57,70)(53,64,58,69)(54,63,59,68)(55,62,60,67) );`

`G=PermutationGroup([[(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,51,46,56),(42,52,47,57),(43,53,48,58),(44,54,49,59),(45,55,50,60),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75)], [(1,59),(2,60),(3,56),(4,57),(5,58),(6,51),(7,52),(8,53),(9,54),(10,55),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,66),(32,67),(33,68),(34,69),(35,70),(36,61),(37,62),(38,63),(39,64),(40,65)], [(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80)], [(1,79,9,74),(2,80,10,75),(3,76,6,71),(4,77,7,72),(5,78,8,73),(11,61,16,66),(12,62,17,67),(13,63,18,68),(14,64,19,69),(15,65,20,70),(21,56,26,51),(22,57,27,52),(23,58,28,53),(24,59,29,54),(25,60,30,55),(31,41,36,46),(32,42,37,47),(33,43,38,48),(34,44,39,49),(35,45,40,50)], [(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),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80)], [(1,23,9,28),(2,22,10,27),(3,21,6,26),(4,25,7,30),(5,24,8,29),(11,31,16,36),(12,35,17,40),(13,34,18,39),(14,33,19,38),(15,32,20,37),(41,76,46,71),(42,80,47,75),(43,79,48,74),(44,78,49,73),(45,77,50,72),(51,61,56,66),(52,65,57,70),(53,64,58,69),(54,63,59,68),(55,62,60,67)]])`

50 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 10A 10B 10C ··· 10L 20A ··· 20T order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 10 10 10 ··· 10 20 ··· 20 size 1 1 2 4 4 2 2 4 4 4 4 20 20 40 2 2 40 40 2 2 4 ··· 4 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 8 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 C5⋊D4 C5⋊D4 C5⋊D4 D4.10D4 2- 1+4.2D5 kernel 2- 1+4.2D5 C20.10D4 D4⋊2Dic5 Dic5⋊Q8 D4.9D10 C5×2- 1+4 C2×C20 C5×D4 C5×Q8 2- 1+4 C2×Q8 C4○D4 C2×C4 D4 Q8 C5 C1 # reps 1 1 2 1 2 1 2 2 2 2 2 4 8 8 8 2 2

Matrix representation of 2- 1+4.2D5 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 1 1 1 2 0 0 0 40 40 40
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 40 40 39 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 1 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 40 40 40 39 0 0 1 0 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 11 12 22 0 0 1 1 31 2 0 0 1 30 0 0 0 0 40 0 40 29
,
 3 34 0 0 0 0 34 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 34 0 0 0 0 7 38 0 0 0 0 0 0 30 40 0 0 0 0 40 11 0 0 0 0 11 11 12 22 0 0 0 30 40 29

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1,0,0,0,1,0,1,40,0,0,0,0,1,40,0,0,0,0,2,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,40,0,40,1,0,0,40,40,0,1,0,0,39,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,40,1,0,0,1,0,40,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,1,1,40,0,0,11,1,30,0,0,0,12,31,0,40,0,0,22,2,0,29],[3,34,0,0,0,0,34,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,7,0,0,0,0,34,38,0,0,0,0,0,0,30,40,11,0,0,0,40,11,11,30,0,0,0,0,12,40,0,0,0,0,22,29] >;`

2- 1+4.2D5 in GAP, Magma, Sage, TeX

`2_-^{1+4}._2D_5`
`% in TeX`

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

`G:=SmallGroup(320,873);`
`// by ID`

`G=gap.SmallGroup(320,873);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,184,570,1684,851,438,102,12550]);`
`// 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=f*a*f^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=a*b,d*c*d^-1=f*c*f^-1=a^2*c,c*e=e*c,d*e=e*d,f*d*f^-1=a^2*c*d,f*e*f^-1=e^-1>;`
`// generators/relations`

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