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## G = D20.4D4order 320 = 26·5

### 4th non-split extension by D20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.4D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — Q8.10D10 — D20.4D4
 Lower central C5 — C10 — C2×C20 — D20.4D4
 Upper central C1 — C2 — C2×C4 — C4.10D4

Generators and relations for D20.4D4
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a15, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, dbd-1=a5b, dcd-1=a15c3 >

Subgroups: 622 in 142 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8, D4.10D4, C40⋊C2, Dic20, C4×Dic5, C10.D4, C5×M4(2), C2×Dic10, C4○D20, C4○D20, Q8×D5, Q82D5, Q8×C10, D207C4, C5×C4.10D4, C8.D10, Dic5⋊Q8, Q8.10D10, D20.4D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, D4.10D4, C2×D20, D4×D5, C22⋊D20, D20.4D4

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 8 type + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D20 D4.10D4 D4×D5 D20.4D4 kernel D20.4D4 D20⋊7C4 C5×C4.10D4 C8.D10 Dic5⋊Q8 Q8.10D10 Dic10 D20 C2×C20 C4.10D4 M4(2) C2×Q8 C2×C4 C5 C4 C1 # reps 1 2 1 2 1 1 2 2 2 2 4 2 8 2 4 2

Matrix representation of D20.4D4 in GL6(𝔽41)

 34 1 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 40 40 1 4 0 0 0 21 20 40
,
 40 7 0 0 0 0 0 1 0 0 0 0 0 0 1 1 40 37 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 21 20 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 1 1 40 37 0 0 0 1 0 0 0 0 21 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 11 11 29 38 0 0 1 1 10 37 0 0 30 40 0 0 0 0 6 21 20 29

`G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,40,0,0,0,1,0,40,21,0,0,0,0,1,20,0,0,0,0,4,40],[40,0,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,40,21,0,0,40,40,0,20,0,0,37,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,21,0,0,0,1,1,0,0,0,1,40,0,0,0,0,0,37,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,1,30,6,0,0,11,1,40,21,0,0,29,10,0,20,0,0,38,37,0,29] >;`

D20.4D4 in GAP, Magma, Sage, TeX

`D_{20}._4D_4`
`% in TeX`

`G:=Group("D20.4D4");`
`// GroupNames label`

`G:=SmallGroup(320,379);`
`// by ID`

`G=gap.SmallGroup(320,379);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,58,1123,570,136,1684,438,12550]);`
`// Polycyclic`

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

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