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G = C2×C10order 20 = 22·5

Abelian group of type [2,10]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C10, SmallGroup(20,5)

Series: Derived Chief Lower central Upper central

C1 — C2×C10
C1C5C10 — C2×C10
C1 — C2×C10
C1 — C2×C10

Generators and relations for C2×C10
 G = < a,b | a2=b10=1, ab=ba >


Character table of C2×C10

 class 12A2B2C5A5B5C5D10A10B10C10D10E10F10G10H10I10J10K10L
 size 11111111111111111111
ρ111111111111111111111    trivial
ρ21-11-11111-111-1-1-1-11-1-1-11    linear of order 2
ρ311-1-11111-1-1-11111-1-1-1-1-1    linear of order 2
ρ41-1-1111111-1-1-1-1-1-1-1111-1    linear of order 2
ρ51111ζ54ζ5ζ52ζ53ζ54ζ54ζ5ζ5ζ52ζ53ζ54ζ52ζ5ζ52ζ53ζ53    linear of order 5
ρ61-11-1ζ54ζ5ζ52ζ5354ζ54ζ55525354ζ5255253ζ53    linear of order 10
ρ711-1-1ζ54ζ5ζ52ζ5354545ζ5ζ52ζ53ζ54525525353    linear of order 10
ρ81-1-11ζ54ζ5ζ52ζ53ζ54545552535452ζ5ζ52ζ5353    linear of order 10
ρ91111ζ53ζ52ζ54ζ5ζ53ζ53ζ52ζ52ζ54ζ5ζ53ζ54ζ52ζ54ζ5ζ5    linear of order 5
ρ101-11-1ζ53ζ52ζ54ζ553ζ53ζ525254553ζ5452545ζ5    linear of order 10
ρ1111-1-1ζ53ζ52ζ54ζ5535352ζ52ζ54ζ5ζ5354525455    linear of order 10
ρ121-1-11ζ53ζ52ζ54ζ5ζ535352525455354ζ52ζ54ζ55    linear of order 10
ρ131111ζ52ζ53ζ5ζ54ζ52ζ52ζ53ζ53ζ5ζ54ζ52ζ5ζ53ζ5ζ54ζ54    linear of order 5
ρ141-11-1ζ52ζ53ζ5ζ5452ζ52ζ535355452ζ553554ζ54    linear of order 10
ρ1511-1-1ζ52ζ53ζ5ζ54525253ζ53ζ5ζ54ζ5255355454    linear of order 10
ρ161-1-11ζ52ζ53ζ5ζ54ζ52525353554525ζ53ζ5ζ5454    linear of order 10
ρ171111ζ5ζ54ζ53ζ52ζ5ζ5ζ54ζ54ζ53ζ52ζ5ζ53ζ54ζ53ζ52ζ52    linear of order 5
ρ181-11-1ζ5ζ54ζ53ζ525ζ5ζ545453525ζ53545352ζ52    linear of order 10
ρ1911-1-1ζ5ζ54ζ53ζ525554ζ54ζ53ζ52ζ55354535252    linear of order 10
ρ201-1-11ζ5ζ54ζ53ζ52ζ5554545352553ζ54ζ53ζ5252    linear of order 10

Permutation representations of C2×C10
Regular action on 20 points - transitive group 20T3
Generators in S20
(1 17)(2 18)(3 19)(4 20)(5 11)(6 12)(7 13)(8 14)(9 15)(10 16)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

G:=sub<Sym(20)| (1,17)(2,18)(3,19)(4,20)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)>;

G:=Group( (1,17)(2,18)(3,19)(4,20)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20) );

G=PermutationGroup([(1,17),(2,18),(3,19),(4,20),(5,11),(6,12),(7,13),(8,14),(9,15),(10,16)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)])

G:=TransitiveGroup(20,3);

Polynomial with Galois group C2×C10 over ℚ
actionf(x)Disc(f)
20T3x20-x19+x17-x16+x14-x13+x11-x10+x9-x7+x6-x4+x3-x+1310·1118

Matrix representation of C2×C10 in GL2(𝔽11) generated by

100
010
,
100
09
G:=sub<GL(2,GF(11))| [10,0,0,10],[10,0,0,9] >;

C2×C10 in GAP, Magma, Sage, TeX

C_2\times C_{10}
% in TeX

G:=Group("C2xC10");
// GroupNames label

G:=SmallGroup(20,5);
// by ID

G=gap.SmallGroup(20,5);
# by ID

G:=PCGroup([3,-2,-2,-5]);
// Polycyclic

G:=Group<a,b|a^2=b^10=1,a*b=b*a>;
// generators/relations

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