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## G = D20.1D4order 320 = 26·5

### 1st non-split extension by D20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.1D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4⋊6D10 — D20.1D4
 Lower central C5 — C10 — C2×C20 — D20.1D4
 Upper central C1 — C2 — C2×C4 — C4.D4

Generators and relations for D20.1D4
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=a5b, dbd-1=a15b, dcd-1=a15c3 >

Subgroups: 750 in 152 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, D4.9D4, C40⋊C2, Dic20, C4×Dic5, C23.D5, C5×M4(2), C2×Dic10, C4○D20, D4×D5, D42D5, C2×C5⋊D4, D4×C10, D207C4, C5×C4.D4, C8.D10, C20.17D4, D46D10, D20.1D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, D4.9D4, C2×D20, D4×D5, C22⋊D20, D20.1D4

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 5 5 8 8 10 10 10 10 10 10 10 10 20 20 20 20 40 ··· 40 size 1 1 2 4 4 20 20 2 2 20 20 20 20 40 2 2 8 8 2 2 4 4 8 8 8 8 4 4 4 4 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 D4.9D4 D4×D5 D20.1D4 kernel D20.1D4 D20⋊7C4 C5×C4.D4 C8.D10 C20.17D4 D4⋊6D10 Dic10 D20 C22×C10 C4.D4 M4(2) C2×D4 C23 C5 C4 C1 # reps 1 2 1 2 1 1 2 2 2 2 4 2 8 2 4 2

Matrix representation of D20.1D4 in GL8(𝔽41)

 0 1 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 35 1 0 0 0 0 0 0 5 40 0 0 0 0 0 0 0 0 34 36 17 8 0 0 0 0 26 7 25 38 0 0 0 0 7 5 12 37 0 0 0 0 31 15 5 29
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 31 3 40 40 0 0 0 0 13 28 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 11 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 35 1
,
 29 25 38 38 0 0 0 0 16 18 3 0 0 0 0 0 9 18 39 16 0 0 0 0 36 32 39 37 0 0 0 0 0 0 0 0 9 0 24 36 0 0 0 0 24 32 4 6 0 0 0 0 27 2 26 5 0 0 0 0 23 26 29 15
,
 29 25 38 38 0 0 0 0 16 18 3 0 0 0 0 0 9 5 39 16 0 0 0 0 8 4 39 37 0 0 0 0 0 0 0 0 9 0 35 5 0 0 0 0 24 32 26 33 0 0 0 0 27 2 32 0 0 0 0 0 23 26 28 9

`G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,35,5,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,34,26,7,31,0,0,0,0,36,7,5,15,0,0,0,0,17,25,12,5,0,0,0,0,8,38,37,29],[0,1,31,13,0,0,0,0,1,0,3,28,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,0,40,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,35,0,0,0,0,0,0,0,1],[29,16,9,36,0,0,0,0,25,18,18,32,0,0,0,0,38,3,39,39,0,0,0,0,38,0,16,37,0,0,0,0,0,0,0,0,9,24,27,23,0,0,0,0,0,32,2,26,0,0,0,0,24,4,26,29,0,0,0,0,36,6,5,15],[29,16,9,8,0,0,0,0,25,18,5,4,0,0,0,0,38,3,39,39,0,0,0,0,38,0,16,37,0,0,0,0,0,0,0,0,9,24,27,23,0,0,0,0,0,32,2,26,0,0,0,0,35,26,32,28,0,0,0,0,5,33,0,9] >;`

D20.1D4 in GAP, Magma, Sage, TeX

`D_{20}._1D_4`
`% in TeX`

`G:=Group("D20.1D4");`
`// GroupNames label`

`G:=SmallGroup(320,373);`
`// by ID`

`G=gap.SmallGroup(320,373);`
`# by ID`

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

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

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