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## G = D20.39D4order 320 = 26·5

### 9th non-split extension by D20 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.39D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — D4⋊8D10 — D20.39D4
 Lower central C5 — C10 — C2×C20 — D20.39D4
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for D20.39D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a13b, dbd=a18b, dcd=a10c-1 >

Subgroups: 750 in 152 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C42, C22⋊C4, M4(2), M4(2), SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.D4, C4≀C2, C4.4D4, C8.C22, C8.C22, 2+ 1+4, C52C8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×D5, D4.9D4, C4.Dic5, C4×Dic5, D10⋊C4, Q8⋊D5, C5⋊Q16, C5×M4(2), C5×SD16, C5×Q16, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, Q82D5, Q8×C10, C5×C4○D4, C20.46D4, D207C4, D42Dic5, C20.C23, C20.23D4, C5×C8.C22, D48D10, D20.39D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, D4.9D4, D4×D5, C2×C5⋊D4, C23⋊D10, D20.39D4

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

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

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

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

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 20A 20B 20C 20D 20E ··· 20J 40A 40B 40C 40D 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 20 20 20 20 20 ··· 20 40 40 40 40 size 1 1 2 4 20 20 20 2 2 4 8 20 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 D4.9D4 D4×D5 D4×D5 D20.39D4 kernel D20.39D4 C20.46D4 D20⋊7C4 D4⋊2Dic5 C20.C23 C20.23D4 C5×C8.C22 D4⋊8D10 Dic10 D20 C5×D4 C5×Q8 C22×D5 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 1 1 2 2 2 2 2 4 4 2 2 2 2

Matrix representation of D20.39D4 in GL6(𝔽41)

 0 1 0 0 0 0 40 34 0 0 0 0 0 0 0 40 0 9 0 0 1 0 32 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 9 0 0 1 0 32 0 0 0 0 18 0 1 0 0 23 0 40 0
,
 40 0 0 0 0 0 7 1 0 0 0 0 0 0 0 9 21 21 0 0 32 0 20 21 0 0 0 0 32 0 0 0 0 0 0 32
,
 40 0 0 0 0 0 7 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,34,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,32,0,40,0,0,9,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,23,0,0,40,0,18,0,0,0,0,32,0,40,0,0,9,0,1,0],[40,7,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,21,20,32,0,0,0,21,21,0,32],[40,7,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D20.39D4 in GAP, Magma, Sage, TeX

`D_{20}._{39}D_4`
`% in TeX`

`G:=Group("D20.39D4");`
`// GroupNames label`

`G:=SmallGroup(320,829);`
`// by ID`

`G=gap.SmallGroup(320,829);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,184,570,1684,851,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^13*b,d*b*d=a^18*b,d*c*d=a^10*c^-1>;`
`// generators/relations`

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