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## G = D20.D5order 400 = 24·52

### 2nd non-split extension by D20 of D5 acting via D5/C5=C2

Aliases: D20.2D5, C522SD16, Dic102D5, C20.11D10, C4.9D52, C52(Q8⋊D5), (C5×C10).8D4, C52(D4.D5), C527C82C2, (C5×D20).1C2, (C5×Dic10)⋊1C2, C10.8(C5⋊D4), (C5×C20).3C22, C2.4(C522D4), SmallGroup(400,66)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C20 — D20.D5
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C5×D20 — D20.D5
 Lower central C52 — C5×C10 — C5×C20 — D20.D5
 Upper central C1 — C2 — C4

Generators and relations for D20.D5
G = < a,b,c,d | a20=b2=c5=1, d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=c-1 >

Smallest permutation representation of D20.D5
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 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 66)(22 65)(23 64)(24 63)(25 62)(26 61)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 74)(34 73)(35 72)(36 71)(37 70)(38 69)(39 68)(40 67)
(1 9 17 5 13)(2 10 18 6 14)(3 11 19 7 15)(4 12 20 8 16)(21 33 25 37 29)(22 34 26 38 30)(23 35 27 39 31)(24 36 28 40 32)(41 53 45 57 49)(42 54 46 58 50)(43 55 47 59 51)(44 56 48 60 52)(61 69 77 65 73)(62 70 78 66 74)(63 71 79 67 75)(64 72 80 68 76)
(1 35 11 25)(2 26 12 36)(3 37 13 27)(4 28 14 38)(5 39 15 29)(6 30 16 40)(7 21 17 31)(8 32 18 22)(9 23 19 33)(10 34 20 24)(41 78 51 68)(42 69 52 79)(43 80 53 70)(44 71 54 61)(45 62 55 72)(46 73 56 63)(47 64 57 74)(48 75 58 65)(49 66 59 76)(50 77 60 67)

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

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

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

43 conjugacy classes

 class 1 2A 2B 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A ··· 20L 20M 20N 20O 20P order 1 2 2 4 4 5 5 5 5 5 5 5 5 8 8 10 10 10 10 10 10 10 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 20 2 20 2 2 2 2 4 4 4 4 50 50 2 2 2 2 4 4 4 4 20 20 20 20 4 ··· 4 20 20 20 20

43 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + - + + - image C1 C2 C2 C2 D4 D5 D5 SD16 D10 C5⋊D4 D4.D5 Q8⋊D5 D52 C52⋊2D4 D20.D5 kernel D20.D5 C52⋊7C8 C5×Dic10 C5×D20 C5×C10 Dic10 D20 C52 C20 C10 C5 C5 C4 C2 C1 # reps 1 1 1 1 1 2 2 2 4 8 2 2 4 4 8

Matrix representation of D20.D5 in GL6(𝔽41)

 1 39 0 0 0 0 1 40 0 0 0 0 0 0 0 1 0 0 0 0 40 7 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 36 15 0 0 0 0 23 5 0 0 0 0 0 0 20 26 0 0 0 0 2 21 0 0 0 0 0 0 17 40 0 0 0 0 1 24
,
 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 34 40 0 0 0 0 1 0
,
 21 13 0 0 0 0 7 20 0 0 0 0 0 0 17 1 0 0 0 0 40 24 0 0 0 0 0 0 6 39 0 0 0 0 38 35

G:=sub<GL(6,GF(41))| [1,1,0,0,0,0,39,40,0,0,0,0,0,0,0,40,0,0,0,0,1,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[36,23,0,0,0,0,15,5,0,0,0,0,0,0,20,2,0,0,0,0,26,21,0,0,0,0,0,0,17,1,0,0,0,0,40,24],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[21,7,0,0,0,0,13,20,0,0,0,0,0,0,17,40,0,0,0,0,1,24,0,0,0,0,0,0,6,38,0,0,0,0,39,35] >;

D20.D5 in GAP, Magma, Sage, TeX

D_{20}.D_5
% in TeX

G:=Group("D20.D5");
// GroupNames label

G:=SmallGroup(400,66);
// by ID

G=gap.SmallGroup(400,66);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,55,218,116,50,970,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^5=1,d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations

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