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## G = D4.11D20order 320 = 26·5

### 1st non-split extension by D4 of D20 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4.11D20
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — D4⋊8D10 — D4.11D20
 Lower central C5 — C10 — C20 — D4.11D20
 Upper central C1 — C2 — C4○D4 — C8○D4

Generators and relations for D4.11D20
G = < a,b,c,d | a4=b2=1, c20=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c19 >

Subgroups: 1094 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C40, C40, Dic10, Dic10, Dic10, C4×D5, D20, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, D4○SD16, C40⋊C2, C40⋊C2, D40, Dic20, C2×C40, C5×M4(2), C2×Dic10, C2×D20, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, C2×C40⋊C2, D407C2, C8⋊D10, C8.D10, C5×C8○D4, D48D10, D4.10D10, D4.11D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, D20, C22×D5, D4○SD16, C2×D20, C23×D5, C22×D20, D4.11D20

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D20 D20 D4○SD16 D4.11D20 kernel D4.11D20 C2×C40⋊C2 D40⋊7C2 C8⋊D10 C8.D10 C5×C8○D4 D4⋊8D10 D4.10D10 C5×D4 C5×Q8 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C5 C1 # reps 1 3 3 3 3 1 1 1 3 1 2 6 6 2 12 4 2 8

Matrix representation of D4.11D20 in GL4(𝔽41) generated by

 0 0 1 0 0 0 0 1 40 0 0 0 0 40 0 0
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 23 16 0 0 25 12 0 0 0 0 23 16 0 0 25 12
,
 12 14 0 0 16 29 0 0 0 0 12 14 0 0 16 29
`G:=sub<GL(4,GF(41))| [0,0,40,0,0,0,0,40,1,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[23,25,0,0,16,12,0,0,0,0,23,25,0,0,16,12],[12,16,0,0,14,29,0,0,0,0,12,16,0,0,14,29] >;`

D4.11D20 in GAP, Magma, Sage, TeX

`D_4._{11}D_{20}`
`% in TeX`

`G:=Group("D4.11D20");`
`// GroupNames label`

`G:=SmallGroup(320,1423);`
`// by ID`

`G=gap.SmallGroup(320,1423);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,80,1684,102,12550]);`
`// Polycyclic`

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

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