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## G = D20.40D4order 320 = 26·5

### 10th non-split extension by D20 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.40D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4.10D10 — D20.40D4
 Lower central C5 — C10 — C2×C20 — D20.40D4
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for D20.40D4
G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=a10c3 >

Subgroups: 558 in 142 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C42, C4⋊C4, M4(2), M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, C8.C22, 2- 1+4, C52C8, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, D4.10D4, C4.Dic5, C4×Dic5, C10.D4, Q8⋊D5, C5⋊Q16, C5×M4(2), C5×SD16, C5×Q16, C2×Dic10, C2×Dic10, C4○D20, C4○D20, D42D5, Q8×D5, Q8×C10, C5×C4○D4, C4.12D20, D207C4, D42Dic5, C20.C23, Dic5⋊Q8, C5×C8.C22, D4.10D10, D20.40D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, D4.10D4, D4×D5, C2×C5⋊D4, C23⋊D10, D20.40D4

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4 D5 D10 D10 D10 C5⋊D4 C5⋊D4 D4.10D4 D4×D5 D4×D5 D20.40D4 kernel D20.40D4 C4.12D20 D20⋊7C4 D4⋊2Dic5 C20.C23 Dic5⋊Q8 C5×C8.C22 D4.10D10 Dic10 D20 C2×Dic5 C5×D4 C5×Q8 C8.C22 M4(2) C2×Q8 C4○D4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 4 4 2 2 2 2

Matrix representation of D20.40D4 in GL8(𝔽41)

 0 40 0 0 0 0 0 0 1 7 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 7 0 0 0 0 0 0 0 0 40 4 0 0 0 0 0 0 20 1 0 0 0 0 0 0 0 0 40 4 0 0 0 0 0 0 20 1
,
 40 17 24 2 0 0 0 0 24 1 39 17 0 0 0 0 17 39 1 24 0 0 0 0 2 24 17 40 0 0 0 0 0 0 0 0 0 0 4 17 0 0 0 0 0 0 40 37 0 0 0 0 37 24 0 0 0 0 0 0 1 4 0 0
,
 17 40 2 24 0 0 0 0 1 24 17 39 0 0 0 0 39 17 24 1 0 0 0 0 24 2 40 17 0 0 0 0 0 0 0 0 0 0 8 33 0 0 0 0 0 0 3 33 0 0 0 0 4 17 0 0 0 0 0 0 40 37 0 0
,
 39 17 24 1 0 0 0 0 24 2 40 17 0 0 0 0 17 40 2 24 0 0 0 0 1 24 17 39 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0

`G:=sub<GL(8,GF(41))| [0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,40,20,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,40,20,0,0,0,0,0,0,4,1],[40,24,17,2,0,0,0,0,17,1,39,24,0,0,0,0,24,39,1,17,0,0,0,0,2,17,24,40,0,0,0,0,0,0,0,0,0,0,37,1,0,0,0,0,0,0,24,4,0,0,0,0,4,40,0,0,0,0,0,0,17,37,0,0],[17,1,39,24,0,0,0,0,40,24,17,2,0,0,0,0,2,17,24,40,0,0,0,0,24,39,1,17,0,0,0,0,0,0,0,0,0,0,4,40,0,0,0,0,0,0,17,37,0,0,0,0,8,3,0,0,0,0,0,0,33,33,0,0],[39,24,17,1,0,0,0,0,17,2,40,24,0,0,0,0,24,40,2,17,0,0,0,0,1,17,24,39,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

D20.40D4 in GAP, Magma, Sage, TeX

`D_{20}._{40}D_4`
`% in TeX`

`G:=Group("D20.40D4");`
`// GroupNames label`

`G:=SmallGroup(320,832);`
`// by ID`

`G=gap.SmallGroup(320,832);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,184,1123,297,136,851,438,102,12550]);`
`// Polycyclic`

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

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