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G = C2×D20  order 80 = 24·5

Direct product of C2 and D20

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D20, C4⋊2D10, C10⋊1D4, C20⋊2C22, D10⋊1C22, C10.3C23, C22.10D10, C5⋊1(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, C2, C4, C22, C22, C5, C2×C4, D4, C23, D5, C10, C10, C2×D4, C20, D10, D10, C2×C10, D20, C2×C20, C22×D5, C2×D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, D20, C22×D5, C2×D20

Character table of C2×D20

 class 12A2B2C2D2E2F2G4A4B5A5B10A10B10C10D10E10F20A20B20C20D20E20F20G20H
 size 111110101010222222222222222222
ρ111111111111111111111111111    trivial
ρ211-1-11-1-11-11111-1-1-11-1-1111-1-11-1    linear of order 2
ρ31111-1-111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-1-1-11-111-1111-1-1-11-11-1-1-111-11    linear of order 2
ρ511-1-11-11-11-1111-1-1-11-11-1-1-111-11    linear of order 2
ρ6111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ711-1-1-111-1-11111-1-1-11-1-1111-1-11-1    linear of order 2
ρ81111-1-1-1-1111111111111111111    linear of order 2
ρ92-22-200000022-2-2-22-2200000000    orthogonal lifted from D4
ρ102-2-2200000022-222-2-2-200000000    orthogonal lifted from D4
ρ1122-2-200002-2-1+√5/2-1-√5/2-1-√5/21-√5/21+√5/21+√5/2-1+√5/21-√5/2-1-√5/21+√5/21+√5/21-√5/2-1+√5/2-1-√5/21-√5/2-1+√5/2    orthogonal lifted from D10
ρ1222-2-20000-22-1-√5/2-1+√5/2-1+√5/21+√5/21-√5/21-√5/2-1-√5/21+√5/21-√5/2-1+√5/2-1+√5/2-1-√5/21+√5/21-√5/2-1-√5/21+√5/2    orthogonal lifted from D10
ρ132222000022-1-√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
ρ142222000022-1+√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
ρ1522220000-2-2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/21+√5/21+√5/21+√5/21-√5/21-√5/21+√5/21-√5/21-√5/2    orthogonal lifted from D10
ρ1622-2-200002-2-1-√5/2-1+√5/2-1+√5/21+√5/21-√5/21-√5/2-1-√5/21+√5/2-1+√5/21-√5/21-√5/21+√5/2-1-√5/2-1+√5/21+√5/2-1-√5/2    orthogonal lifted from D10
ρ1722220000-2-2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/21-√5/21-√5/21-√5/21+√5/21+√5/21-√5/21+√5/21+√5/2    orthogonal lifted from D10
ρ1822-2-20000-22-1+√5/2-1-√5/2-1-√5/21-√5/21+√5/21+√5/2-1+√5/21-√5/21+√5/2-1-√5/2-1-√5/2-1+√5/21-√5/21+√5/2-1+√5/21-√5/2    orthogonal lifted from D10
ρ192-2-22000000-1+√5/2-1-√5/21+√5/2-1+√5/2-1-√5/21+√5/21-√5/21-√5/2-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5    orthogonal lifted from D20
ρ202-22-2000000-1+√5/2-1-√5/21+√5/21-√5/21+√5/2-1-√5/21-√5/2-1+√5/2-ζ4ζ53+ζ4ζ52-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5-ζ43ζ54+ζ43ζ5ζ4ζ53-ζ4ζ52ζ43ζ54-ζ43ζ5ζ43ζ54-ζ43ζ5    orthogonal lifted from D20
ρ212-22-2000000-1+√5/2-1-√5/21+√5/21-√5/21+√5/2-1-√5/21-√5/2-1+√5/2ζ4ζ53-ζ4ζ52ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5ζ43ζ54-ζ43ζ5-ζ4ζ53+ζ4ζ52-ζ43ζ54+ζ43ζ5-ζ43ζ54+ζ43ζ5    orthogonal lifted from D20
ρ222-22-2000000-1-√5/2-1+√5/21-√5/21+√5/21-√5/2-1+√5/21+√5/2-1-√5/2-ζ43ζ54+ζ43ζ5-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52ζ4ζ53-ζ4ζ52ζ43ζ54-ζ43ζ5-ζ4ζ53+ζ4ζ52-ζ4ζ53+ζ4ζ52    orthogonal lifted from D20
ρ232-2-22000000-1+√5/2-1-√5/21+√5/2-1+√5/2-1-√5/21+√5/21-√5/21-√5/2ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5    orthogonal lifted from D20
ρ242-2-22000000-1-√5/2-1+√5/21-√5/2-1-√5/2-1+√5/21-√5/21+√5/21+√5/2ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52    orthogonal lifted from D20
ρ252-2-22000000-1-√5/2-1+√5/21-√5/2-1-√5/2-1+√5/21-√5/21+√5/21+√5/2-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52    orthogonal lifted from D20
ρ262-22-2000000-1-√5/2-1+√5/21-√5/21+√5/21-√5/2-1+√5/21+√5/2-1-√5/2ζ43ζ54-ζ43ζ5ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52-ζ4ζ53+ζ4ζ52-ζ43ζ54+ζ43ζ5ζ4ζ53-ζ4ζ52ζ4ζ53-ζ4ζ52    orthogonal lifted from D20

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

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

C2×D20 is a maximal subgroup of
 D20⋊6C4  D20⋊5C4  C20.46D4  D10.D4  C20⋊4D4  C4.D20  C22⋊D20  D10⋊D4  D20⋊8C4  D10.13D4  C4⋊D20  C8⋊D10  C20⋊7D4  C20⋊D4  C20.23D4  D4⋊D10  C2×D4×D5  D4⋊8D10
C2×D20 is a maximal quotient of
 C20⋊2Q8  C20⋊4D4  C4.D20  C22⋊D20  C22.D20  C4⋊D20  D10⋊2Q8  D40⋊7C2  C8⋊D10  C8.D10  C20⋊7D4

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

1000
0100
00400
00040
,
0100
40000
00401
00535
,
0100
1000
00400
0051
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
 

Export

Character table of C2×D20 in TeX

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