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G = D20⋊D4order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D105D8, D206D4, (C2×D8)⋊4D5, (C5×D4)⋊5D4, (C2×C8)⋊3D10, (C2×D4)⋊3D10, C2.28(D5×D8), C4.59(D4×D5), (C10×D8)⋊13C2, C202D43C2, C54(C22⋊D8), D42(C5⋊D4), C10.45(C2×D8), C20.46(C2×D4), (C2×C40)⋊27C22, (D4×C10)⋊3C22, D101C827C2, D205C429C2, C10.55C22≀C2, D4⋊Dic528C2, C4⋊Dic519C22, (C2×Dic5).73D4, C22.256(D4×D5), C2.29(D8⋊D5), C10.50(C8⋊C22), (C2×C20).433C23, (C22×D5).127D4, C2.23(C23⋊D10), (C2×D20).120C22, (C2×D4×D5)⋊2C2, (C2×D4⋊D5)⋊19C2, C4.36(C2×C5⋊D4), (C2×C52C8)⋊7C22, (C2×C4×D5).48C22, (C2×C10).346(C2×D4), (C2×C4).523(C22×D5), SmallGroup(320,783)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20⋊D4
C1C5C10C2×C10C2×C20C2×C4×D5C2×D4×D5 — D20⋊D4
C5C10C2×C20 — D20⋊D4
C1C22C2×C4C2×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 [×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 [×4], C10 [×3], C10 [×3], C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×4], C22×C4, C2×D4 [×2], C2×D4 [×7], C24, Dic5 [×2], C20 [×2], D10 [×2], D10 [×14], C2×C10, C2×C10 [×7], C22⋊C8, D4⋊C4 [×2], C4⋊D4, C2×D8, C2×D8, C22×D4, C52C8, C40, C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×Dic5, C5⋊D4 [×6], C2×C20, C5×D4 [×2], C5×D4 [×3], C22×D5, C22×D5 [×9], C22×C10 [×2], C22⋊D8, C2×C52C8, C4⋊Dic5, D4⋊D5 [×2], C23.D5, C2×C40, C5×D8 [×2], C2×C4×D5, C2×D20, D4×D5 [×4], C2×C5⋊D4 [×2], D4×C10 [×2], C23×D5, D101C8, D205C4, D4⋊Dic5, C2×D4⋊D5, C202D4, C10×D8, C2×D4×D5, D20⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, D8 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×D8, C8⋊C22, C5⋊D4 [×2], C22×D5, C22⋊D8, D4×D5 [×2], 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 23)(2 22)(3 21)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 80)(48 79)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)
(1 54 29 64)(2 53 30 63)(3 52 31 62)(4 51 32 61)(5 50 33 80)(6 49 34 79)(7 48 35 78)(8 47 36 77)(9 46 37 76)(10 45 38 75)(11 44 39 74)(12 43 40 73)(13 42 21 72)(14 41 22 71)(15 60 23 70)(16 59 24 69)(17 58 25 68)(18 57 26 67)(19 56 27 66)(20 55 28 65)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 37)(22 26)(23 35)(25 33)(27 31)(28 40)(30 38)(32 36)(41 67)(42 76)(43 65)(44 74)(45 63)(46 72)(47 61)(48 70)(49 79)(50 68)(51 77)(52 66)(53 75)(54 64)(55 73)(56 62)(57 71)(58 80)(59 69)(60 78)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D5A5B8A8B8C8D10A···10F10G···10N20A20B20C20D40A···40H
order12222222222444455888810···1010···102020202040···40
size111144810102020222040224420202···28···844444···4

47 irreducible representations

dim1111111122222222244444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D8D10D10C5⋊D4C8⋊C22D4×D5D4×D5D5×D8D8⋊D5
kernelD20⋊D4D101C8D205C4D4⋊Dic5C2×D4⋊D5C202D4C10×D8C2×D4×D5D20C2×Dic5C5×D4C22×D5C2×D8D10C2×C8C2×D4D4C10C4C22C2C2
# reps1111111121212424812244

Matrix representation of D20⋊D4 in GL6(𝔽41)

150000
16400000
007100
00334000
0000400
0000040
,
4000000
2510000
00404000
000100
0000400
0000231
,
0220000
2800000
00343500
008700
00003923
0000392
,
100000
010000
00343500
008700
000010
00001840

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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