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G = D4⋊D20order 320 = 26·5

1st semidirect product of D4 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D202D4, D42D20, D104D8, C4⋊C41D10, (C5×D4)⋊1D4, (C2×C8)⋊2D10, C2.8(D5×D8), (C2×D40)⋊4C2, C4.84(D4×D5), C4.1(C2×D20), D4⋊C44D5, C4⋊D201C2, C52(C22⋊D8), (C2×C40)⋊2C22, C10.22(C2×D8), D101C84C2, D206C46C2, C20.107(C2×D4), C10.19C22≀C2, (C2×D4).134D10, (C2×D20)⋊12C22, (C2×Dic5).26D4, C22.171(D4×D5), C2.10(D40⋊C2), C10.55(C8⋊C22), (C2×C20).213C23, (C22×D5).107D4, (D4×C10).34C22, C2.22(C22⋊D20), (C2×D4×D5)⋊1C2, (C2×D4⋊D5)⋊2C2, (C5×C4⋊C4)⋊3C22, (C5×D4⋊C4)⋊4C2, (C2×C52C8)⋊2C22, (C2×C4×D5).11C22, (C2×C10).226(C2×D4), (C2×C4).320(C22×D5), SmallGroup(320,400)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D4⋊D20
C1C5C10C2×C10C2×C20C2×C4×D5C2×D4×D5 — D4⋊D20
C5C10C2×C20 — D4⋊D20
C1C22C2×C4D4⋊C4

Generators and relations for D4⋊D20
 G = < a,b,c,d | a4=b2=c20=d2=1, bab=cac-1=dad=a-1, cbc-1=dbd=a-1b, dcd=c-1 >

Subgroups: 1166 in 198 conjugacy classes, 45 normal (37 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×2], C22, C22 [×23], C5, C8 [×2], C2×C4, C2×C4 [×4], D4 [×2], D4 [×12], C23 [×12], D5 [×5], C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×4], C22×C4, C2×D4, C2×D4 [×8], C24, Dic5, C20 [×2], C20, D10 [×2], D10 [×17], C2×C10, C2×C10 [×4], C22⋊C8, D4⋊C4, D4⋊C4, C4⋊D4, C2×D8 [×2], C22×D4, C52C8, C40, C4×D5 [×2], D20 [×2], D20 [×5], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×D5, C22×D5 [×10], C22×C10, C22⋊D8, D40 [×2], C2×C52C8, D10⋊C4, D4⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20 [×2], C2×D20, D4×D5 [×4], C2×C5⋊D4, D4×C10, C23×D5, D206C4, D101C8, C5×D4⋊C4, C4⋊D20, C2×D40, C2×D4⋊D5, C2×D4×D5, D4⋊D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, D8 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×D8, C8⋊C22, D20 [×2], C22×D5, C22⋊D8, C2×D20, D4×D5 [×2], C22⋊D20, D5×D8, D40⋊C2, D4⋊D20

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222222222444455888810···1010101010202020202020202040···40
size111144101020204022820224420202···28888444488884···4

47 irreducible representations

dim11111111222222222244444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D8D10D10D10D20C8⋊C22D4×D5D4×D5D5×D8D40⋊C2
kernelD4⋊D20D206C4D101C8C5×D4⋊C4C4⋊D20C2×D40C2×D4⋊D5C2×D4×D5D20C2×Dic5C5×D4C22×D5D4⋊C4D10C4⋊C4C2×C8C2×D4D4C10C4C22C2C2
# reps11111111212124222812244

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

1000
0100
00040
0010
,
40000
04000
0010
00040
,
251100
143900
001212
001229
,
393000
4200
002929
002912
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[25,14,0,0,11,39,0,0,0,0,12,12,0,0,12,29],[39,4,0,0,30,2,0,0,0,0,29,29,0,0,29,12] >;

D4⋊D20 in GAP, Magma, Sage, TeX

D_4\rtimes D_{20}
% in TeX

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

G:=SmallGroup(320,400);
// by ID

G=gap.SmallGroup(320,400);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,851,438,102,12550]);
// Polycyclic

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

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