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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, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C4×D4, C4×D4, C22≀C2, C4⋊D4, C41D4, C22×D4, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, D42, C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20, C2×D20, D4×D5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C4×D20, C204D4, C22⋊D20, C4⋊D20, C207D4, D4×C20, C22×D20, C2×D4×D5, D4×D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, D20, C22×D5, D42, C2×D20, D4×D5, C23×D5, C22×D20, C2×D4×D5, D48D10, D4×D20

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

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

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

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

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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