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G = D44D20order 320 = 26·5

3rd semidirect product of D4 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D44D20, Q84D20, D2015D4, C422D10, Dic1015D4, M4(2)⋊3D10, C4≀C21D5, (C5×D4)⋊3D4, (C5×Q8)⋊3D4, C4.9(C2×D20), C8⋊D108C2, C204D46C2, C52(D44D4), C4○D4.1D10, C4.125(D4×D5), D204C45C2, D4⋊D101C2, D48D101C2, (C4×C20)⋊11C22, C20.337(C2×D4), (C22×D5).2D4, C22.29(D4×D5), C10.27C22≀C2, C20.46D41C2, (C2×D20)⋊13C22, C4.Dic54C22, (C2×C20).262C23, C4○D20.11C22, C2.30(C22⋊D20), (C5×M4(2))⋊10C22, (C5×C4≀C2)⋊1C2, (C2×C10).26(C2×D4), (C5×C4○D4).3C22, (C2×C4).109(C22×D5), SmallGroup(320,449)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D44D20
C1C5C10C2×C10C2×C20C2×D20D48D10 — D44D20
C5C10C2×C20 — D44D20
C1C2C2×C4C4≀C2

Generators and relations for D44D20
 G = < a,b,c,d | a4=b2=c20=d2=1, bab=dad=a-1, ac=ca, cbc-1=a-1b, dbd=ab, dcd=c-1 >

Subgroups: 974 in 168 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×4], C22, C22 [×11], C5, C8 [×2], C2×C4, C2×C4 [×5], D4, D4 [×15], Q8, Q8, C23 [×5], D5 [×4], C10, C10 [×2], C42, M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4 [×8], C4○D4, C4○D4 [×3], Dic5, C20 [×2], C20 [×3], D10 [×10], C2×C10, C2×C10, C4.D4, C4≀C2, C4≀C2, C41D4, C8⋊C22 [×2], 2+ 1+4, C52C8, C40, Dic10, C4×D5 [×3], D20, D20 [×10], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C22×D5 [×2], C22×D5 [×3], D44D4, C40⋊C2, D40, C4.Dic5, D4⋊D5, Q8⋊D5, C4×C20, C5×M4(2), C2×D20 [×2], C2×D20 [×3], C4○D20, C4○D20, D4×D5 [×3], Q82D5, C5×C4○D4, D204C4, C20.46D4, C5×C4≀C2, C204D4, C8⋊D10, D4⋊D10, D48D10, D44D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, D44D4, C2×D20, D4×D5 [×2], C22⋊D20, D44D20

Smallest permutation representation of D44D20
On 40 points
Generators in S40
(1 20 8 12)(2 16 9 13)(3 17 10 14)(4 18 6 15)(5 19 7 11)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)
(1 22)(2 38)(3 34)(4 30)(5 26)(6 40)(7 36)(8 32)(9 28)(10 24)(11 21)(12 37)(13 33)(14 29)(15 25)(16 23)(17 39)(18 35)(19 31)(20 27)
(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)
(1 2)(3 5)(7 10)(8 9)(11 17)(12 16)(13 20)(14 19)(15 18)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)

G:=sub<Sym(40)| (1,20,8,12)(2,16,9,13)(3,17,10,14)(4,18,6,15)(5,19,7,11)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30), (1,22)(2,38)(3,34)(4,30)(5,26)(6,40)(7,36)(8,32)(9,28)(10,24)(11,21)(12,37)(13,33)(14,29)(15,25)(16,23)(17,39)(18,35)(19,31)(20,27), (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), (1,2)(3,5)(7,10)(8,9)(11,17)(12,16)(13,20)(14,19)(15,18)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)>;

G:=Group( (1,20,8,12)(2,16,9,13)(3,17,10,14)(4,18,6,15)(5,19,7,11)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30), (1,22)(2,38)(3,34)(4,30)(5,26)(6,40)(7,36)(8,32)(9,28)(10,24)(11,21)(12,37)(13,33)(14,29)(15,25)(16,23)(17,39)(18,35)(19,31)(20,27), (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), (1,2)(3,5)(7,10)(8,9)(11,17)(12,16)(13,20)(14,19)(15,18)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33) );

G=PermutationGroup([(1,20,8,12),(2,16,9,13),(3,17,10,14),(4,18,6,15),(5,19,7,11),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30)], [(1,22),(2,38),(3,34),(4,30),(5,26),(6,40),(7,36),(8,32),(9,28),(10,24),(11,21),(12,37),(13,33),(14,29),(15,25),(16,23),(17,39),(18,35),(19,31),(20,27)], [(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)], [(1,2),(3,5),(7,10),(8,9),(11,17),(12,16),(13,20),(14,19),(15,18),(21,24),(22,23),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33)])

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A8B10A10B10C10D10E10F20A20B20C20D20E···20N20O20P40A40B40C40D
order1222222244444455881010101010102020202020···20202040404040
size11242020204022444202284022448822224···4888888

44 irreducible representations

dim11111111222222222224444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D4D5D10D10D10D20D20D44D4D4×D5D4×D5D44D20
kernelD44D20D204C4C20.46D4C5×C4≀C2C204D4C8⋊D10D4⋊D10D48D10Dic10D20C5×D4C5×Q8C22×D5C4≀C2C42M4(2)C4○D4D4Q8C5C4C22C1
# reps11111111111122222442228

Matrix representation of D44D20 in GL4(𝔽41) generated by

30900
321100
001132
00930
,
001132
00930
30900
321100
,
34100
40000
002730
001132
,
13400
04000
001114
00930
G:=sub<GL(4,GF(41))| [30,32,0,0,9,11,0,0,0,0,11,9,0,0,32,30],[0,0,30,32,0,0,9,11,11,9,0,0,32,30,0,0],[34,40,0,0,1,0,0,0,0,0,27,11,0,0,30,32],[1,0,0,0,34,40,0,0,0,0,11,9,0,0,14,30] >;

D44D20 in GAP, Magma, Sage, TeX

D_4\rtimes_4D_{20}
% in TeX

G:=Group("D4:4D20");
// GroupNames label

G:=SmallGroup(320,449);
// by ID

G=gap.SmallGroup(320,449);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,570,1684,851,102,12550]);
// Polycyclic

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

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