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## G = D20⋊D4order 320 = 26·5

### 6th semidirect product of D20 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20⋊D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C2×D4×D5 — D20⋊D4
 Lower central C5 — C10 — C2×C20 — D20⋊D4
 Upper central C1 — C22 — C2×C4 — C2×D8

Generators and relations for D20⋊D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=cac-1=a-1, dad=a9, cbc-1=a3b, dbd=a8b, dcd=c-1 >

Subgroups: 1070 in 198 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C2×D8, C22×D4, C52C8, C40, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×D5, C22×D5, C22×C10, C22⋊D8, C2×C52C8, C4⋊Dic5, D4⋊D5, C23.D5, C2×C40, C5×D8, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, D101C8, D205C4, D4⋊Dic5, C2×D4⋊D5, C202D4, C10×D8, C2×D4×D5, D20⋊D4
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C22≀C2, C2×D8, C8⋊C22, C5⋊D4, C22×D5, C22⋊D8, D4×D5, C2×C5⋊D4, D5×D8, D8⋊D5, C23⋊D10, D20⋊D4

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D5 D8 D10 D10 C5⋊D4 C8⋊C22 D4×D5 D4×D5 D5×D8 D8⋊D5 kernel D20⋊D4 D10⋊1C8 D20⋊5C4 D4⋊Dic5 C2×D4⋊D5 C20⋊2D4 C10×D8 C2×D4×D5 D20 C2×Dic5 C5×D4 C22×D5 C2×D8 D10 C2×C8 C2×D4 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 2 1 2 4 2 4 8 1 2 2 4 4

Matrix representation of D20⋊D4 in GL6(𝔽41)

 1 5 0 0 0 0 16 40 0 0 0 0 0 0 7 1 0 0 0 0 33 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 25 1 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 23 1
,
 0 22 0 0 0 0 28 0 0 0 0 0 0 0 34 35 0 0 0 0 8 7 0 0 0 0 0 0 39 23 0 0 0 0 39 2
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 35 0 0 0 0 8 7 0 0 0 0 0 0 1 0 0 0 0 0 18 40

G:=sub<GL(6,GF(41))| [1,16,0,0,0,0,5,40,0,0,0,0,0,0,7,33,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,25,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,40,23,0,0,0,0,0,1],[0,28,0,0,0,0,22,0,0,0,0,0,0,0,34,8,0,0,0,0,35,7,0,0,0,0,0,0,39,39,0,0,0,0,23,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,18,0,0,0,0,0,40] >;

D20⋊D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes D_4
% in TeX

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

G:=SmallGroup(320,783);
// by ID

G=gap.SmallGroup(320,783);
# by ID

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

G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^9,c*b*c^-1=a^3*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations

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