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## G = D4.9D20order 320 = 26·5

### 4th non-split extension by D4 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4.9D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4.10D10 — D4.9D20
 Lower central C5 — C10 — C2×C20 — D4.9D20
 Upper central C1 — C2 — C2×C4 — C4≀C2

Generators and relations for D4.9D20
G = < a,b,c,d | a4=b2=1, c20=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=c19 >

Subgroups: 590 in 142 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C42, C4⋊C4, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.10D4, C4≀C2, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C52C8, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, D4.10D4, C40⋊C2, Dic20, C4.Dic5, C4⋊Dic5, D4.D5, C5⋊Q16, C4×C20, C5×M4(2), C2×Dic10, C2×Dic10, C4○D20, C4○D20, D42D5, Q8×D5, C5×C4○D4, D204C4, C4.12D20, C5×C4≀C2, C202Q8, C8.D10, D4.9D10, D4.10D10, D4.9D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, D4.10D4, C2×D20, D4×D5, C22⋊D20, D4.9D20

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

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

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

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

44 conjugacy classes

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

44 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 4 type + + + + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4 D5 D10 D10 D10 D20 D20 D4.10D4 D4×D5 D4×D5 D4.9D20 kernel D4.9D20 D20⋊4C4 C4.12D20 C5×C4≀C2 C20⋊2Q8 C8.D10 D4.9D10 D4.10D10 Dic10 D20 C2×Dic5 C5×D4 C5×Q8 C4≀C2 C42 M4(2) C4○D4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 4 4 2 2 2 8

Matrix representation of D4.9D20 in GL4(𝔽41) generated by

 11 32 0 0 9 30 0 0 0 0 30 9 0 0 32 11
,
 0 0 40 0 0 0 0 40 40 0 0 0 0 40 0 0
,
 0 0 34 1 0 0 40 0 27 30 0 0 11 32 0 0
,
 19 22 0 0 32 22 0 0 0 0 3 3 0 0 24 38
`G:=sub<GL(4,GF(41))| [11,9,0,0,32,30,0,0,0,0,30,32,0,0,9,11],[0,0,40,0,0,0,0,40,40,0,0,0,0,40,0,0],[0,0,27,11,0,0,30,32,34,40,0,0,1,0,0,0],[19,32,0,0,22,22,0,0,0,0,3,24,0,0,3,38] >;`

D4.9D20 in GAP, Magma, Sage, TeX

`D_4._9D_{20}`
`% in TeX`

`G:=Group("D4.9D20");`
`// GroupNames label`

`G:=SmallGroup(320,453);`
`// by ID`

`G=gap.SmallGroup(320,453);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,58,1123,136,851,438,102,12550]);`
`// Polycyclic`

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

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