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G = D201D4order 320 = 26·5

1st semidirect product of D20 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D201D4, Dic101D4, C23.7D20, M4(2)⋊2D10, C4.79(D4×D5), C20⋊D41C2, C8⋊D105C2, C51(D44D4), C20.92(C2×D4), C4.D42D5, D207C42C2, D46D102C2, (C2×D4).14D10, (C2×C20).4C23, C10.16C22≀C2, (C2×D20)⋊11C22, (C4×Dic5)⋊2C22, C4○D20.2C22, (C22×C10).20D4, C22.11(C2×D20), (D4×C10).14C22, (C5×M4(2))⋊9C22, C2.19(C22⋊D20), (C5×C4.D4)⋊4C2, (C2×C10).21(C2×D4), (C2×C4).4(C22×D5), SmallGroup(320,374)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D201D4
C1C5C10C20C2×C20C4○D20D46D10 — D201D4
C5C10C2×C20 — D201D4
C1C2C2×C4C4.D4

Generators and relations for D201D4
 G = < a,b,c,d | a20=b2=c4=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a13b, dbd=a3b, dcd=c-1 >

Subgroups: 942 in 168 conjugacy classes, 39 normal (17 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 [×3], C10, C10 [×3], C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×D4 [×7], C4○D4 [×4], Dic5 [×4], C20 [×2], D10 [×7], C2×C10, C2×C10 [×4], C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], D20 [×2], C2×Dic5 [×3], C5⋊D4 [×10], C2×C20, C5×D4 [×2], C22×D5 [×3], C22×C10 [×2], D44D4, C40⋊C2 [×2], D40 [×2], C4×Dic5, C5×M4(2) [×2], C2×D20, C4○D20 [×2], D4×D5 [×2], D42D5 [×2], C2×C5⋊D4 [×4], D4×C10, D207C4 [×2], C5×C4.D4, C8⋊D10 [×2], C20⋊D4, D46D10, D201D4
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, D201D4

Smallest permutation representation of D201D4
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)
(1 6 11 16)(2 15 12 5)(3 4 13 14)(7 20 17 10)(8 9 18 19)(22 30)(23 39)(24 28)(25 37)(27 35)(29 33)(32 40)(34 38)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A8B10A10B10C10D10E10F10G10H20A20B20C20D40A···40H
order12222222444444558810101010101010102020202040···40
size11244202040222020202022882244888844448···8

38 irreducible representations

dim1111112222222448
type++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5D10D10D20D44D4D4×D5D201D4
kernelD201D4D207C4C5×C4.D4C8⋊D10C20⋊D4D46D10Dic10D20C22×C10C4.D4M4(2)C2×D4C23C5C4C1
# reps1212112222428242

Matrix representation of D201D4 in GL6(𝔽41)

6400000
3610000
0004000
001000
002828405
0003161
,
2110000
37390000
000010
002828405
001000
003232013
,
35340000
560000
000100
0040000
0000400
00038040
,
35340000
560000
0040000
000100
002828405
003001

G:=sub<GL(6,GF(41))| [6,36,0,0,0,0,40,1,0,0,0,0,0,0,0,1,28,0,0,0,40,0,28,3,0,0,0,0,40,16,0,0,0,0,5,1],[2,37,0,0,0,0,11,39,0,0,0,0,0,0,0,28,1,32,0,0,0,28,0,32,0,0,1,40,0,0,0,0,0,5,0,13],[35,5,0,0,0,0,34,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,38,0,0,0,0,40,0,0,0,0,0,0,40],[35,5,0,0,0,0,34,6,0,0,0,0,0,0,40,0,28,3,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,5,1] >;

D201D4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_1D_4
% in TeX

G:=Group("D20:1D4");
// GroupNames label

G:=SmallGroup(320,374);
// by ID

G=gap.SmallGroup(320,374);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,226,1123,570,136,1684,438,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^13*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

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