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## G = D20.14D4order 320 = 26·5

### 14th non-split extension by D20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.14D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — Q8.10D10 — D20.14D4
 Lower central C5 — C10 — C2×C20 — D20.14D4
 Upper central C1 — C2 — C2×C4 — C4.4D4

Generators and relations for D20.14D4
G = < a,b,c,d | a20=b2=d2=1, c4=a10, bab=cac-1=a-1, dad=a9, cbc-1=a3b, dbd=a18b, dcd=a10c3 >

Subgroups: 590 in 146 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C52C8, Dic10, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, D4.8D4, C4.Dic5, D4⋊D5, D4.D5, C4×C20, C5×C22⋊C4, C4○D20, C4○D20, Q8×D5, Q82D5, D4×C10, Q8×C10, D204C4, C20.10D4, D4.D10, C5×C4.4D4, Q8.10D10, D20.14D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, D4.8D4, D4×D5, C2×C5⋊D4, C23⋊D10, D20.14D4

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 C5⋊D4 D4.8D4 D4×D5 D20.14D4 kernel D20.14D4 D20⋊4C4 C20.10D4 D4.D10 C5×C4.4D4 Q8.10D10 Dic10 D20 C2×C20 C4.4D4 C42 C2×D4 C2×Q8 C2×C4 C5 C4 C1 # reps 1 2 1 2 1 1 2 2 2 2 2 2 2 8 2 4 8

Matrix representation of D20.14D4 in GL6(𝔽41)

 0 1 0 0 0 0 40 34 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 20 20 1 4 0 0 31 0 20 40
,
 30 32 0 0 0 0 27 11 0 0 0 0 0 0 0 0 32 0 0 0 25 25 32 5 0 0 9 0 0 0 0 0 6 31 8 16
,
 14 30 0 0 0 0 14 27 0 0 0 0 0 0 20 20 1 4 0 0 0 0 40 0 0 0 1 0 0 0 0 0 33 13 0 21
,
 27 11 0 0 0 0 27 14 0 0 0 0 0 0 21 21 40 37 0 0 0 0 40 0 0 0 0 40 0 0 0 0 28 8 10 20

`G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,34,0,0,0,0,0,0,0,1,20,31,0,0,40,0,20,0,0,0,0,0,1,20,0,0,0,0,4,40],[30,27,0,0,0,0,32,11,0,0,0,0,0,0,0,25,9,6,0,0,0,25,0,31,0,0,32,32,0,8,0,0,0,5,0,16],[14,14,0,0,0,0,30,27,0,0,0,0,0,0,20,0,1,33,0,0,20,0,0,13,0,0,1,40,0,0,0,0,4,0,0,21],[27,27,0,0,0,0,11,14,0,0,0,0,0,0,21,0,0,28,0,0,21,0,40,8,0,0,40,40,0,10,0,0,37,0,0,20] >;`

D20.14D4 in GAP, Magma, Sage, TeX

`D_{20}._{14}D_4`
`% in TeX`

`G:=Group("D20.14D4");`
`// GroupNames label`

`G:=SmallGroup(320,689);`
`// by ID`

`G=gap.SmallGroup(320,689);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,1123,570,297,136,1684,12550]);`
`// 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=a^10*c^3>;`
`// generators/relations`

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