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G = D20.2D4order 320 = 26·5

2nd non-split extension by D20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.2D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D20.2C4 — D20.2D4
 Lower central C5 — C10 — C2×C20 — D20.2D4
 Upper central C1 — C2 — C2×C4 — C4.D4

Generators and relations for D20.2D4
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, dbd-1=a15b, dcd-1=a15c3 >

Subgroups: 414 in 104 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C52C8, C52C8, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×C10, D4.3D4, C8×D5, C8⋊D5, C40⋊C2, Dic20, C2×C52C8, C4.Dic5, D4⋊D5, D4.D5, C5×M4(2), C2×Dic10, C4○D20, D4×C10, C20.53D4, C4.12D20, C5×C4.D4, D20.2C4, C8.D10, D4.D10, C2×D4.D5, D20.2D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C22×D5, D4.3D4, C4○D20, D4×D5, D10⋊D4, D20.2D4

Smallest permutation representation of D20.2D4
On 80 points
Generators in S80
```(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 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 76)(22 75)(23 74)(24 73)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 80)(38 79)(39 78)(40 77)
(1 27 6 22 11 37 16 32)(2 38 7 33 12 28 17 23)(3 29 8 24 13 39 18 34)(4 40 9 35 14 30 19 25)(5 31 10 26 15 21 20 36)(41 61 56 66 51 71 46 76)(42 72 57 77 52 62 47 67)(43 63 58 68 53 73 48 78)(44 74 59 79 54 64 49 69)(45 65 60 70 55 75 50 80)
(1 37 6 22 11 27 16 32)(2 38 7 23 12 28 17 33)(3 39 8 24 13 29 18 34)(4 40 9 25 14 30 19 35)(5 21 10 26 15 31 20 36)(41 66 46 71 51 76 56 61)(42 67 47 72 52 77 57 62)(43 68 48 73 53 78 58 63)(44 69 49 74 54 79 59 64)(45 70 50 75 55 80 60 65)```

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 8 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 C4○D20 D4.3D4 D4×D5 D20.2D4 kernel D20.2D4 C20.53D4 C4.12D20 C5×C4.D4 D20.2C4 C8.D10 D4.D10 C2×D4.D5 C5⋊2C8 Dic10 D20 C4.D4 C2×C10 M4(2) C2×D4 C22 C5 C4 C1 # reps 1 1 1 1 1 1 1 1 2 1 1 2 2 4 2 8 2 4 2

Matrix representation of D20.2D4 in GL6(𝔽41)

 40 1 0 0 0 0 33 7 0 0 0 0 0 0 1 9 0 0 0 0 18 40 0 0 0 0 5 4 0 40 0 0 36 0 1 0
,
 32 25 0 0 0 0 5 9 0 0 0 0 0 0 0 15 11 30 0 0 17 27 0 7 0 0 37 40 26 12 0 0 34 6 26 29
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 5 39 0 0 0 0 9 23 18 0 0 0 0 36 4 0 0 1 0 4 36
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 5 39 0 0 0 18 40 23 23 0 0 4 16 36 37 0 0 36 21 4 5

`G:=sub<GL(6,GF(41))| [40,33,0,0,0,0,1,7,0,0,0,0,0,0,1,18,5,36,0,0,9,40,4,0,0,0,0,0,0,1,0,0,0,0,40,0],[32,5,0,0,0,0,25,9,0,0,0,0,0,0,0,17,37,34,0,0,15,27,40,6,0,0,11,0,26,26,0,0,30,7,12,29],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,5,9,0,0,0,0,39,23,36,4,0,0,0,18,4,36],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,18,4,36,0,0,5,40,16,21,0,0,39,23,36,4,0,0,0,23,37,5] >;`

D20.2D4 in GAP, Magma, Sage, TeX

`D_{20}._2D_4`
`% in TeX`

`G:=Group("D20.2D4");`
`// GroupNames label`

`G:=SmallGroup(320,375);`
`// by ID`

`G=gap.SmallGroup(320,375);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,297,136,1684,851,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^3>;`
`// generators/relations`

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