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G = C2×D20order 80 = 24·5

Direct product of C2 and D20

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

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

C1C10 — C2×D20
C1C5C10D10C22×D5 — C2×D20
C5C10 — C2×D20
C1C22C2×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 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+5/21-5/21-5/24ζ534ζ52ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5    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+5/24ζ534ζ524ζ534ζ52ζ4ζ534ζ5243ζ5443ζ543ζ5443ζ5ζ4ζ534ζ52ζ43ζ5443ζ5ζ43ζ5443ζ5    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+5/2ζ4ζ534ζ52ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ5ζ43ζ5443ζ54ζ534ζ5243ζ5443ζ543ζ5443ζ5    orthogonal lifted from D20
ρ222-22-2000000-1-5/2-1+5/21-5/21+5/21-5/2-1+5/21+5/2-1-5/243ζ5443ζ543ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ52ζ4ζ534ζ52ζ43ζ5443ζ54ζ534ζ524ζ534ζ52    orthogonal lifted from D20
ρ232-2-22000000-1+5/2-1-5/21+5/2-1+5/2-1-5/21+5/21-5/21-5/2ζ4ζ534ζ524ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ54ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5    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+5/2ζ43ζ5443ζ543ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ5243ζ5443ζ54ζ534ζ52ζ4ζ534ζ52    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+5/243ζ5443ζ5ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ4ζ534ζ52ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ52    orthogonal lifted from D20
ρ262-22-2000000-1-5/2-1+5/21-5/21+5/21-5/2-1+5/21+5/2-1-5/2ζ43ζ5443ζ5ζ43ζ5443ζ543ζ5443ζ54ζ534ζ524ζ534ζ5243ζ5443ζ5ζ4ζ534ζ52ζ4ζ534ζ52    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

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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