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G = D2023D4order 320 = 26·5

1st semidirect product of D20 and D4 acting through Inn(D20)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2023D4, C4214D10, C10.1032+ 1+4, C4⋊C447D10, (C4×D4)⋊12D5, (D4×C20)⋊14C2, (C4×D20)⋊28C2, C52(D45D4), C4.139(D4×D5), D10⋊D47C2, C207D418C2, (C4×C20)⋊19C22, C22⋊C446D10, D10.37(C2×D4), C20.345(C2×D4), (C22×D20)⋊9C2, (C22×C4)⋊12D10, C23⋊D1020C2, (C2×D4).213D10, C4.D2016C2, C223(C4○D20), (C2×C10).94C24, C4⋊Dic559C22, C10.49(C22×D4), D10.13D47C2, C20.48D410C2, (C2×C20).782C23, (C22×C20)⋊16C22, C10.D43C22, C2.15(D48D10), D10⋊C430C22, C23.94(C22×D5), (C2×Dic10)⋊53C22, (C2×D20).218C22, (D4×C10).305C22, (C2×Dic5).40C23, (C23×D5).39C22, C22.119(C23×D5), C23.D5.11C22, (C22×C10).164C23, (C22×D5).182C23, C2.22(C2×D4×D5), (C2×C4○D20)⋊7C2, (C2×C4×D5)⋊48C22, (C2×C10)⋊2(C4○D4), (C5×C4⋊C4)⋊59C22, (D5×C22⋊C4)⋊28C2, C2.45(C2×C4○D20), C10.41(C2×C4○D4), (C2×C5⋊D4)⋊3C22, (C5×C22⋊C4)⋊56C22, (C2×C4).158(C22×D5), SmallGroup(320,1222)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D2023D4
C1C5C10C2×C10C22×D5C23×D5D5×C22⋊C4 — D2023D4
C5C2×C10 — D2023D4
C1C22C4×D4

Generators and relations for D2023D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd=c-1 >

Subgroups: 1462 in 334 conjugacy classes, 107 normal (51 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×27], C5, C2×C4 [×5], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], D5 [×6], C10 [×3], C10 [×3], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×4], D10 [×18], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C22⋊C4 [×2], C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×6], D20 [×4], D20 [×6], C2×Dic5 [×4], C5⋊D4 [×6], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×D5 [×4], C22×D5 [×10], C22×C10 [×2], D45D4, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×8], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×4], C2×D20 [×2], C2×D20 [×2], C2×D20 [×4], C4○D20 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C23×D5 [×2], C4×D20, C4.D20, D5×C22⋊C4 [×2], D10⋊D4 [×2], D10.13D4 [×2], C20.48D4, C207D4, C23⋊D10 [×2], D4×C20, C22×D20, C2×C4○D20, D2023D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ 1+4, C22×D5 [×7], D45D4, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D48D10, D2023D4

Smallest permutation representation of D2023D4
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 22)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)
(1 27 49 74)(2 28 50 75)(3 29 51 76)(4 30 52 77)(5 31 53 78)(6 32 54 79)(7 33 55 80)(8 34 56 61)(9 35 57 62)(10 36 58 63)(11 37 59 64)(12 38 60 65)(13 39 41 66)(14 40 42 67)(15 21 43 68)(16 22 44 69)(17 23 45 70)(18 24 46 71)(19 25 47 72)(20 26 48 73)
(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L4A···4F4G4H4I4J4K4L5A5B10A···10F10G···10N20A···20H20I···20X
order12222222222224···44444445510···1010···1020···2020···20
size11112241010101020202···24420202020222···24···42···24···4

65 irreducible representations

dim111111111111222222222444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10D10D10C4○D202+ 1+4D4×D5D48D10
kernelD2023D4C4×D20C4.D20D5×C22⋊C4D10⋊D4D10.13D4C20.48D4C207D4C23⋊D10D4×C20C22×D20C2×C4○D20D20C4×D4C2×C10C42C22⋊C4C4⋊C4C22×C4C2×D4C22C10C4C2
# reps1112221121114242424216144

Matrix representation of D2023D4 in GL4(𝔽41) generated by

1000
0100
001411
00309
,
1000
0100
003027
003211
,
04000
1000
00171
004024
,
1000
04000
0010
0001
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,14,30,0,0,11,9],[1,0,0,0,0,1,0,0,0,0,30,32,0,0,27,11],[0,1,0,0,40,0,0,0,0,0,17,40,0,0,1,24],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1] >;

D2023D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{23}D_4
% in TeX

G:=Group("D20:23D4");
// GroupNames label

G:=SmallGroup(320,1222);
// by ID

G=gap.SmallGroup(320,1222);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,675,570,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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