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## G = D4.3D20order 320 = 26·5

### 3rd non-split extension by D4 of D20 acting via D20/C20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4.3D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C2×C40⋊C2 — D4.3D20
 Lower central C5 — C10 — C2×C20 — D4.3D20
 Upper central C1 — C2 — C2×C4 — C8○D4

Generators and relations for D4.3D20
G = < a,b,c,d | a4=b2=1, c20=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c19 >

Subgroups: 446 in 104 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C52C8, C40, C40, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, D4.3D4, C40⋊C2, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C2×Dic10, C2×D20, C5×C4○D4, C40.6C4, C20.46D4, C4.12D20, C2×C40⋊C2, D4⋊D10, D4.9D10, C5×C8○D4, D4.3D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, D20, C5⋊D4, C22×D5, D4.3D4, C2×D20, C4○D20, C2×C5⋊D4, C207D4, D4.3D20

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 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 4 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 2 2 4 40 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 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 D20 D20 C4○D20 D4.3D4 D4.3D20 kernel D4.3D20 C40.6C4 C20.46D4 C4.12D20 C2×C40⋊C2 D4⋊D10 D4.9D10 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 1 1 1 1 1 1 2 1 1 2 2 2 2 2 8 4 4 8 2 8

Matrix representation of D4.3D20 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 39 40 0 0 0 0 0 0 40 40 0 0 0 0 2 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 40 0 0 0 0 2 1 0 0 1 1 0 0 0 0 39 40 0 0
,
 19 27 0 0 0 0 27 19 0 0 0 0 0 0 11 26 0 0 0 0 30 0 0 0 0 0 0 0 11 26 0 0 0 0 30 0
,
 9 30 0 0 0 0 11 32 0 0 0 0 0 0 0 26 0 0 0 0 11 0 0 0 0 0 0 0 30 15 0 0 0 0 11 11

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,39,0,0,0,0,1,40,0,0,0,0,0,0,40,2,0,0,0,0,40,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,39,0,0,0,0,1,40,0,0,40,2,0,0,0,0,40,1,0,0],[19,27,0,0,0,0,27,19,0,0,0,0,0,0,11,30,0,0,0,0,26,0,0,0,0,0,0,0,11,30,0,0,0,0,26,0],[9,11,0,0,0,0,30,32,0,0,0,0,0,0,0,11,0,0,0,0,26,0,0,0,0,0,0,0,30,11,0,0,0,0,15,11] >;`

D4.3D20 in GAP, Magma, Sage, TeX

`D_4._3D_{20}`
`% in TeX`

`G:=Group("D4.3D20");`
`// GroupNames label`

`G:=SmallGroup(320,768);`
`// by ID`

`G=gap.SmallGroup(320,768);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,1123,297,136,1684,102,12550]);`
`// Polycyclic`

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

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