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## G = D20.31D4order 320 = 26·5

### 1st non-split extension by D20 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.31D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C22×D20 — D20.31D4
 Lower central C5 — C10 — C2×C20 — D20.31D4
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for D20.31D4
G = < a,b,c,d | a20=b2=d2=1, c4=a10, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd=a15c3 >

Subgroups: 1070 in 188 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C40, Dic10, D20, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×C10, C22⋊SD16, C40⋊C2, C10.D4, C4⋊Dic5, C23.D5, C2×C40, C2×Dic10, C2×D20, C2×D20, C22×C20, C23×D5, D205C4, C5×C22⋊C8, C2×C40⋊C2, C20.48D4, C22×D20, D20.31D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C22≀C2, C2×SD16, C8⋊C22, D20, C22×D5, C22⋊SD16, C40⋊C2, C2×D20, D4×D5, C22⋊D20, C2×C40⋊C2, C8⋊D10, D20.31D4

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

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

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

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

59 conjugacy classes

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

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 SD16 D10 D10 D20 D20 C40⋊C2 C8⋊C22 D4×D5 C8⋊D10 kernel D20.31D4 D20⋊5C4 C5×C22⋊C8 C2×C40⋊C2 C20.48D4 C22×D20 D20 C2×C20 C22×C10 C22⋊C8 C2×C10 C2×C8 C22×C4 C2×C4 C23 C22 C10 C4 C2 # reps 1 2 1 2 1 1 4 1 1 2 4 4 2 4 4 16 1 4 4

Matrix representation of D20.31D4 in GL6(𝔽41)

 10 5 0 0 0 0 29 31 0 0 0 0 0 0 35 1 0 0 0 0 5 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 10 5 0 0 0 0 13 31 0 0 0 0 0 0 40 40 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 24 1
,
 29 7 0 0 0 0 16 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 22 36 0 0 0 0 31 19
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 17 40

`G:=sub<GL(6,GF(41))| [10,29,0,0,0,0,5,31,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[10,13,0,0,0,0,5,31,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,40,24,0,0,0,0,0,1],[29,16,0,0,0,0,7,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,22,31,0,0,0,0,36,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,40] >;`

D20.31D4 in GAP, Magma, Sage, TeX

`D_{20}._{31}D_4`
`% in TeX`

`G:=Group("D20.31D4");`
`// GroupNames label`

`G:=SmallGroup(320,358);`
`// by ID`

`G=gap.SmallGroup(320,358);`
`# by ID`

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

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

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