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## G = C2×D20order 80 = 24·5

### Direct product of C2 and D20

Aliases: C2×D20, C42D10, C101D4, C202C22, D101C22, C10.3C23, C22.10D10, C51(C2×D4), (C2×C4)⋊2D5, (C2×C20)⋊3C2, (C22×D5)⋊1C2, C2.4(C22×D5), (C2×C10).10C22, SmallGroup(80,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×D20
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C2×D20
 Lower central C5 — C10 — C2×D20
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D20
G = < a,b,c | a2=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 178 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C22, C22 [×8], C5, C2×C4, D4 [×4], C23 [×2], D5 [×4], C10, C10 [×2], C2×D4, C20 [×2], D10 [×4], D10 [×4], C2×C10, D20 [×4], C2×C20, C22×D5 [×2], C2×D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], D20 [×2], C22×D5, C2×D20

Character table of C2×D20

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 10A 10B 10C 10D 10E 10F 20A 20B 20C 20D 20E 20F 20G 20H size 1 1 1 1 10 10 10 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 linear of order 2 ρ3 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 linear of order 2 ρ5 1 1 -1 -1 1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 linear of order 2 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 linear of order 2 ρ8 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ9 2 -2 2 -2 0 0 0 0 0 0 2 2 -2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 0 0 0 0 0 2 2 -2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 0 0 0 0 2 -2 -1+√5/2 -1-√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 -1+√5/2 1-√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ12 2 2 -2 -2 0 0 0 0 -2 2 -1-√5/2 -1+√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 -1-√5/2 1+√5/2 1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 -1-√5/2 1+√5/2 orthogonal lifted from D10 ρ13 2 2 2 2 0 0 0 0 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ14 2 2 2 2 0 0 0 0 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ15 2 2 2 2 0 0 0 0 -2 -2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ16 2 2 -2 -2 0 0 0 0 2 -2 -1-√5/2 -1+√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 -1-√5/2 1+√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ17 2 2 2 2 0 0 0 0 -2 -2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ18 2 2 -2 -2 0 0 0 0 -2 2 -1+√5/2 -1-√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 -1+√5/2 1-√5/2 1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 -1+√5/2 1-√5/2 orthogonal lifted from D10 ρ19 2 -2 -2 2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 orthogonal lifted from D20 ρ20 2 -2 2 -2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1+√5/2 -1-√5/2 1-√5/2 -1+√5/2 -ζ4ζ53+ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 ζ43ζ54-ζ43ζ5 orthogonal lifted from D20 ρ21 2 -2 2 -2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1+√5/2 -1-√5/2 1-√5/2 -1+√5/2 ζ4ζ53-ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 -ζ43ζ54+ζ43ζ5 orthogonal lifted from D20 ρ22 2 -2 2 -2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1-√5/2 -1+√5/2 1+√5/2 -1-√5/2 -ζ43ζ54+ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 -ζ4ζ53+ζ4ζ52 orthogonal lifted from D20 ρ23 2 -2 -2 2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 orthogonal lifted from D20 ρ24 2 -2 -2 2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 orthogonal lifted from D20 ρ25 2 -2 -2 2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 orthogonal lifted from D20 ρ26 2 -2 2 -2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1-√5/2 -1+√5/2 1+√5/2 -1-√5/2 ζ43ζ54-ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 -ζ4ζ53+ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ4ζ53-ζ4ζ52 ζ4ζ53-ζ4ζ52 orthogonal lifted from D20

Smallest permutation representation of C2×D20
On 40 points
Generators in S40
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)
(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 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)

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

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

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

C2×D20 is a maximal subgroup of
D206C4  D205C4  C20.46D4  D10.D4  C204D4  C4.D20  C22⋊D20  D10⋊D4  D208C4  D10.13D4  C4⋊D20  C8⋊D10  C207D4  C20⋊D4  C20.23D4  D4⋊D10  C2×D4×D5  D48D10
C2×D20 is a maximal quotient of
C202Q8  C204D4  C4.D20  C22⋊D20  C22.D20  C4⋊D20  D102Q8  D407C2  C8⋊D10  C8.D10  C207D4

Matrix representation of C2×D20 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 0 1 0 0 40 0 0 0 0 0 40 1 0 0 5 35
,
 0 1 0 0 1 0 0 0 0 0 40 0 0 0 5 1
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,40,0,0,1,0,0,0,0,0,40,5,0,0,1,35],[0,1,0,0,1,0,0,0,0,0,40,5,0,0,0,1] >;

C2×D20 in GAP, Magma, Sage, TeX

C_2\times D_{20}
% in TeX

G:=Group("C2xD20");
// GroupNames label

G:=SmallGroup(80,37);
// by ID

G=gap.SmallGroup(80,37);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,182,42,1604]);
// Polycyclic

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

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