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## G = 2+ 1+4.2D5order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — 2+ 1+4.2D5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C23.18D10 — 2+ 1+4.2D5
 Lower central C5 — C10 — C22×C10 — 2+ 1+4.2D5
 Upper central C1 — C2 — C23 — 2+ 1+4

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

Subgroups: 526 in 160 conjugacy classes, 43 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4○D4, Dic5, C20, C2×C10, C2×C10, C23⋊C4, C22.D4, 2+ 1+4, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C22×C10, C23.7D4, C10.D4, C23.D5, C23.D5, C22×Dic5, D4×C10, D4×C10, C5×C4○D4, C23⋊Dic5, C23.18D10, C5×2+ 1+4, 2+ 1+4.2D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, C23.7D4, C2×C5⋊D4, C242D5, 2+ 1+4.2D5

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 8 type + + + + + + + + - image C1 C2 C2 C2 D4 D4 D5 D10 C5⋊D4 C5⋊D4 C23.7D4 2+ 1+4.2D5 kernel 2+ 1+4.2D5 C23⋊Dic5 C23.18D10 C5×2+ 1+4 C2×C20 C22×C10 2+ 1+4 C2×D4 C2×C4 C23 C5 C1 # reps 1 3 3 1 3 3 2 6 12 12 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 1 0 40 40 0 0 0 0 1 0 0 0 0 40 0 0 0 0 2 1 40 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 1 1 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 40 0 0 0 0 0 40 0 0 0 0 0 0 9 9 32 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 9
,
 1 40 0 0 0 0 0 40 0 0 0 0 0 0 9 9 32 0 0 0 23 32 9 9 0 0 0 0 0 9 0 0 0 0 32 0
,
 16 1 0 0 0 0 0 18 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
,
 25 40 0 0 0 0 9 16 0 0 0 0 0 0 9 0 0 0 0 0 23 32 9 9 0 0 0 0 9 0 0 0 0 0 0 9

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,2,0,0,0,0,40,1,0,0,40,1,0,40,0,0,40,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,1,0,1,0,0,0,1,0,0,1],[1,0,0,0,0,0,40,40,0,0,0,0,0,0,9,0,0,0,0,0,9,32,0,0,0,0,32,0,32,0,0,0,0,0,0,9],[1,0,0,0,0,0,40,40,0,0,0,0,0,0,9,23,0,0,0,0,9,32,0,0,0,0,32,9,0,32,0,0,0,9,9,0],[16,0,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,9,0,0,0,0,40,16,0,0,0,0,0,0,9,23,0,0,0,0,0,32,0,0,0,0,0,9,9,0,0,0,0,9,0,9] >;`

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(320,870);`
`// by ID`

`G=gap.SmallGroup(320,870);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,570,438,12550]);`
`// Polycyclic`

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

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