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## G = D20.38D4order 320 = 26·5

### 8th non-split extension by D20 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.38D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4.10D10 — D20.38D4
 Lower central C5 — C10 — C2×C20 — D20.38D4
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D20.38D4
G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=c3 >

Subgroups: 590 in 146 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), D8, SD16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 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, C22×C10, D4.8D4, C4.Dic5, C4×Dic5, D4⋊D5, D4.D5, C23.D5, C5×M4(2), C5×D8, C5×SD16, C2×Dic10, C2×Dic10, C4○D20, C4○D20, D42D5, Q8×D5, D4×C10, C5×C4○D4, C4.12D20, D207C4, D42Dic5, D4.D10, C20.17D4, C5×C8⋊C22, D4.10D10, D20.38D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, D4.8D4, D4×D5, C2×C5⋊D4, C23⋊D10, D20.38D4

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D ··· 4H 5A 5B 8A 8B 10A 10B 10C 10D 10E ··· 10J 20A 20B 20C 20D 20E 20F 40A 40B 40C 40D order 1 2 2 2 2 2 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 8 20 2 2 4 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 D4.8D4 D4×D5 D4×D5 D20.38D4 kernel D20.38D4 C4.12D20 D20⋊7C4 D4⋊2Dic5 D4.D10 C20.17D4 C5×C8⋊C22 D4.10D10 Dic10 D20 C2×Dic5 C5×D4 C5×Q8 C8⋊C22 M4(2) C2×D4 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 2

Matrix representation of D20.38D4 in GL6(𝔽41)

 25 0 0 0 0 0 0 23 0 0 0 0 0 0 32 0 0 0 0 0 5 9 0 0 0 0 0 0 32 0 0 0 9 0 9 9
,
 0 23 0 0 0 0 25 0 0 0 0 0 0 0 0 16 32 23 0 0 0 32 0 5 0 0 9 16 0 0 0 0 0 25 0 9
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 20 40 39 0 0 0 40 0 37 0 0 32 25 0 0 0 0 9 39 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 37 0 0 40 0 0 0 0 0 0 21 0 1

`G:=sub<GL(6,GF(41))| [25,0,0,0,0,0,0,23,0,0,0,0,0,0,32,5,0,9,0,0,0,9,0,0,0,0,0,0,32,9,0,0,0,0,0,9],[0,25,0,0,0,0,23,0,0,0,0,0,0,0,0,0,9,0,0,0,16,32,16,25,0,0,32,0,0,0,0,0,23,5,0,9],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,9,0,0,20,40,25,39,0,0,40,0,0,0,0,0,39,37,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,21,0,0,1,0,0,0,0,0,0,37,0,1] >;`

D20.38D4 in GAP, Magma, Sage, TeX

`D_{20}._{38}D_4`
`% in TeX`

`G:=Group("D20.38D4");`
`// GroupNames label`

`G:=SmallGroup(320,828);`
`// by ID`

`G=gap.SmallGroup(320,828);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,1123,297,136,851,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=1,c^4=d^2=a^10,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

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