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## G = D4×D20order 320 = 26·5

### Direct product of D4 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D4×D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — D4×D20
 Lower central C5 — C2×C10 — D4×D20
 Upper central C1 — C22 — C4×D4

Generators and relations for D4×D20
G = < a,b,c,d | a4=b2=c20=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 2086 in 428 conjugacy classes, 123 normal (29 characteristic)
C1, C2 [×3], C2 [×12], C4 [×4], C4 [×5], C22, C22 [×4], C22 [×40], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×4], D4 [×30], C23 [×2], C23 [×26], D5 [×8], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4, C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×31], C24 [×4], Dic5 [×2], C20 [×4], C20 [×3], D10 [×4], D10 [×32], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×D4, C4×D4, C22≀C2 [×4], C4⋊D4 [×4], C41D4, C22×D4 [×4], C4×D5 [×4], D20 [×4], D20 [×18], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×D5 [×6], C22×D5 [×20], C22×C10 [×2], D42, C4⋊Dic5, D10⋊C4 [×6], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, C2×D20 [×10], C2×D20 [×8], D4×D5 [×8], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C23×D5 [×4], C4×D20, C204D4, C22⋊D20 [×4], C4⋊D20 [×2], C207D4 [×2], D4×C20, C22×D20 [×2], C2×D4×D5 [×2], D4×D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C24, D10 [×7], C22×D4 [×2], 2+ 1+4, D20 [×4], C22×D5 [×7], D42, C2×D20 [×6], D4×D5 [×2], C23×D5, C22×D20, C2×D4×D5, D48D10, D4×D20

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

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D10 D10 D20 2+ 1+4 D4×D5 D4⋊8D10 kernel D4×D20 C4×D20 C20⋊4D4 C22⋊D20 C4⋊D20 C20⋊7D4 D4×C20 C22×D20 C2×D4×D5 D20 C5×D4 C4×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C10 C4 C2 # reps 1 1 1 4 2 2 1 2 2 4 4 2 2 4 2 4 2 16 1 4 4

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

 40 0 0 0 0 40 0 0 0 0 1 37 0 0 21 40
,
 1 0 0 0 0 1 0 0 0 0 1 37 0 0 0 40
,
 2 30 0 0 27 16 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 5 1 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,21,0,0,37,40],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,37,40],[2,27,0,0,30,16,0,0,0,0,40,0,0,0,0,40],[40,5,0,0,0,1,0,0,0,0,1,0,0,0,0,1] >;

D4×D20 in GAP, Magma, Sage, TeX

D_4\times D_{20}
% in TeX

G:=Group("D4xD20");
// GroupNames label

G:=SmallGroup(320,1221);
// by ID

G=gap.SmallGroup(320,1221);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,80,12550]);
// Polycyclic

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

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