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## G = D20.15D4order 320 = 26·5

### 15th non-split extension by D20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.15D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — Q8.10D10 — D20.15D4
 Lower central C5 — C10 — C2×C20 — D20.15D4
 Upper central C1 — C2 — C2×C4 — C4⋊Q8

Generators and relations for D20.15D4
G = < a,b,c,d | a20=b2=d2=1, c4=a10, bab=cac-1=a-1, dad=a9, cbc-1=a3b, dbd=a18b, dcd=c3 >

Subgroups: 558 in 142 conjugacy classes, 39 normal (19 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, C52C8, Dic10, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, D4.10D4, C4.Dic5, Q8⋊D5, C5⋊Q16, C4×C20, C5×C4⋊C4, C4○D20, C4○D20, Q8×D5, Q82D5, Q8×C10, D204C4, C20.10D4, C20.C23, C5×C4⋊Q8, Q8.10D10, D20.15D4
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, D20.15D4

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 C5⋊D4 D4.10D4 D4×D5 D20.15D4 kernel D20.15D4 D20⋊4C4 C20.10D4 C20.C23 C5×C4⋊Q8 Q8.10D10 Dic10 D20 C2×C20 C4⋊Q8 C42 C2×Q8 C2×C4 C5 C4 C1 # reps 1 2 1 2 1 1 2 2 2 2 2 4 8 2 4 8

Matrix representation of D20.15D4 in GL4(𝔽41) generated by

 0 37 0 0 4 0 0 0 0 0 0 10 0 0 31 0
,
 0 0 29 24 0 0 24 12 28 26 0 0 26 13 0 0
,
 0 0 34 14 0 0 14 7 27 34 0 0 34 14 0 0
,
 0 0 0 1 0 0 40 0 0 40 0 0 1 0 0 0
`G:=sub<GL(4,GF(41))| [0,4,0,0,37,0,0,0,0,0,0,31,0,0,10,0],[0,0,28,26,0,0,26,13,29,24,0,0,24,12,0,0],[0,0,27,34,0,0,34,14,34,14,0,0,14,7,0,0],[0,0,0,1,0,0,40,0,0,40,0,0,1,0,0,0] >;`

D20.15D4 in GAP, Magma, Sage, TeX

`D_{20}._{15}D_4`
`% in TeX`

`G:=Group("D20.15D4");`
`// GroupNames label`

`G:=SmallGroup(320,722);`
`// by ID`

`G=gap.SmallGroup(320,722);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,184,1123,570,297,136,1684,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=d^2=1,c^4=a^10,b*a*b=c*a*c^-1=a^-1,d*a*d=a^9,c*b*c^-1=a^3*b,d*b*d=a^18*b,d*c*d=c^3>;`
`// generators/relations`

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