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G = D205D4order 320 = 26·5

5th semidirect product of D20 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D205D4, C426D10, Dic105D4, (C2×D4)⋊2D10, C41D44D5, C4.55(D4×D5), C53(D44D4), C20.35(C2×D4), D46D104C2, (C4×C20)⋊14C22, C20.D46C2, (D4×C10)⋊2C22, D204C413C2, C10.53C22≀C2, D4.D103C2, (C22×C10).23D4, C4.Dic57C22, (C2×C20).395C23, C4○D20.21C22, C23.11(C5⋊D4), C2.21(C23⋊D10), (C5×C41D4)⋊4C2, (C2×C10).526(C2×D4), C22.33(C2×C5⋊D4), (C2×C4).118(C22×D5), SmallGroup(320,704)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D205D4
C1C5C10C20C2×C20C4○D20D46D10 — D205D4
C5C10C2×C20 — D205D4
C1C2C2×C4C41D4

Generators and relations for D205D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a5b, dbd=a10b, dcd=c-1 >

Subgroups: 750 in 168 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×4], C22, C22 [×11], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×16], Q8 [×2], C23 [×2], C23 [×3], D5 [×2], C10, C10 [×4], C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4 [×2], C2×D4 [×6], C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×7], C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, C52C8 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20, C2×C20, C5×D4 [×8], C22×D5 [×2], C22×C10 [×2], C22×C10, D44D4, C4.Dic5 [×2], D4⋊D5 [×2], D4.D5 [×2], C4×C20, C4○D20 [×2], D4×D5 [×2], D42D5 [×2], C2×C5⋊D4 [×2], D4×C10 [×2], D4×C10 [×2], D204C4 [×2], C20.D4, D4.D10 [×2], C5×C41D4, D46D10, D205D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D44D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, D205D4

Smallest permutation representation of D205D4
On 40 points
Generators in S40
(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 25)(2 24)(3 23)(4 22)(5 21)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 6)(2 17)(3 8)(4 19)(5 10)(7 12)(9 14)(11 16)(13 18)(15 20)(21 26)(22 37)(23 28)(24 39)(25 30)(27 32)(29 34)(31 36)(33 38)(35 40)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A8B10A···10F10G···10N20A···20L
order12222222444444558810···1010···1020···20
size1124482020224420202240402···28···84···4

44 irreducible representations

dim1111112222222444
type++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5D10D10C5⋊D4D44D4D4×D5D205D4
kernelD205D4D204C4C20.D4D4.D10C5×C41D4D46D10Dic10D20C22×C10C41D4C42C2×D4C23C5C4C1
# reps1212112222248248

Matrix representation of D205D4 in GL4(𝔽41) generated by

233600
231800
2337025
2237160
,
1090
102525
00400
1823400
,
13900
14000
1823400
1823040
,
13900
04000
02301
02310
G:=sub<GL(4,GF(41))| [23,23,23,22,36,18,37,37,0,0,0,16,0,0,25,0],[1,1,0,18,0,0,0,23,9,25,40,40,0,25,0,0],[1,1,18,18,39,40,23,23,0,0,40,0,0,0,0,40],[1,0,0,0,39,40,23,23,0,0,0,1,0,0,1,0] >;

D205D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_5D_4
% in TeX

G:=Group("D20:5D4");
// GroupNames label

G:=SmallGroup(320,704);
// by ID

G=gap.SmallGroup(320,704);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,1123,570,297,136,1684,12550]);
// 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,d*a*d=a^11,c*b*c^-1=a^5*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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