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G = D4.4D20order 320 = 26·5

4th non-split extension by D4 of D20 acting via D20/C20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4.4D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C2×D40 — D4.4D20
 Lower central C5 — C10 — C2×C20 — D4.4D20
 Upper central C1 — C2 — C2×C4 — C8○D4

Generators and relations for D4.4D20
G = < a,b,c,d | a4=b2=d2=1, c20=a2, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=a2c19 >

Subgroups: 542 in 108 conjugacy classes, 39 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C52C8, C40, C40, D20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, D4.4D4, D40, C4.Dic5, D4⋊D5, Q8⋊D5, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C2×D20, C5×C4○D4, C40.6C4, C20.46D4, C2×D40, D4⋊D10, C5×C8○D4, D4.4D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, D20, C5⋊D4, C22×D5, D4.4D4, C2×D20, C4○D20, C2×C5⋊D4, C207D4, D4.4D20

Smallest permutation representation of D4.4D20
On 80 points
Generators in S80
```(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 71 61 51)(42 72 62 52)(43 73 63 53)(44 74 64 54)(45 75 65 55)(46 76 66 56)(47 77 67 57)(48 78 68 58)(49 79 69 59)(50 80 70 60)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)
(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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 46)(42 45)(43 44)(47 80)(48 79)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 D20 D20 C4○D20 D4.4D4 D4.4D20 kernel D4.4D20 C40.6C4 C20.46D4 C2×D40 D4⋊D10 C5×C8○D4 C40 C5×D4 C5×Q8 C8○D4 C2×C10 C2×C8 M4(2) C4○D4 C8 D4 Q8 C22 C5 C1 # reps 1 1 2 1 2 1 2 1 1 2 2 2 2 2 8 4 4 8 2 8

Matrix representation of D4.4D20 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 1 1 40 37 0 0 0 20 21 1
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 40 37 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 21 20 40
,
 23 0 0 0 0 0 0 25 0 0 0 0 0 0 12 29 0 0 0 0 12 12 0 0 0 0 29 29 24 7 0 0 6 0 35 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 12 29 0 0 0 0 29 29 24 7 0 0 12 6 35 17

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,1,0,0,0,40,0,1,20,0,0,0,0,40,21,0,0,0,0,37,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,40,21,0,0,40,40,0,20,0,0,37,0,0,40],[23,0,0,0,0,0,0,25,0,0,0,0,0,0,12,12,29,6,0,0,29,12,29,0,0,0,0,0,24,35,0,0,0,0,7,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,29,12,0,0,12,29,29,6,0,0,0,0,24,35,0,0,0,0,7,17] >;`

D4.4D20 in GAP, Magma, Sage, TeX

`D_4._4D_{20}`
`% in TeX`

`G:=Group("D4.4D20");`
`// GroupNames label`

`G:=SmallGroup(320,769);`
`// by ID`

`G=gap.SmallGroup(320,769);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,1123,297,136,1684,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^20=a^2,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=a^2*c^19>;`
`// generators/relations`

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