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G = C22order 22 = 2·11

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C22, also denoted Z22, SmallGroup(22,2)

Series: Derived Chief Lower central Upper central

C1 — C22
C1C11 — C22
C1 — C22
C1 — C22

Generators and relations for C22
 G = < a | a22=1 >


Character table of C22

 class 1211A11B11C11D11E11F11G11H11I11J22A22B22C22D22E22F22G22H22I22J
 size 1111111111111111111111
ρ11111111111111111111111    trivial
ρ21-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311ζ112ζ113ζ114ζ115ζ116ζ117ζ118ζ119ζ1110ζ11ζ1110ζ112ζ113ζ114ζ115ζ116ζ117ζ118ζ119ζ11    linear of order 11
ρ41-1ζ112ζ113ζ114ζ115ζ116ζ117ζ118ζ119ζ1110ζ11111011211311411511611711811911    linear of order 22 faithful
ρ511ζ114ζ116ζ118ζ1110ζ11ζ113ζ115ζ117ζ119ζ112ζ119ζ114ζ116ζ118ζ1110ζ11ζ113ζ115ζ117ζ112    linear of order 11
ρ61-1ζ114ζ116ζ118ζ1110ζ11ζ113ζ115ζ117ζ119ζ112119114116118111011113115117112    linear of order 22 faithful
ρ711ζ116ζ119ζ11ζ114ζ117ζ1110ζ112ζ115ζ118ζ113ζ118ζ116ζ119ζ11ζ114ζ117ζ1110ζ112ζ115ζ113    linear of order 11
ρ81-1ζ116ζ119ζ11ζ114ζ117ζ1110ζ112ζ115ζ118ζ113118116119111141171110112115113    linear of order 22 faithful
ρ911ζ118ζ11ζ115ζ119ζ112ζ116ζ1110ζ113ζ117ζ114ζ117ζ118ζ11ζ115ζ119ζ112ζ116ζ1110ζ113ζ114    linear of order 11
ρ101-1ζ118ζ11ζ115ζ119ζ112ζ116ζ1110ζ113ζ117ζ114117118111151191121161110113114    linear of order 22 faithful
ρ1111ζ1110ζ114ζ119ζ113ζ118ζ112ζ117ζ11ζ116ζ115ζ116ζ1110ζ114ζ119ζ113ζ118ζ112ζ117ζ11ζ115    linear of order 11
ρ121-1ζ1110ζ114ζ119ζ113ζ118ζ112ζ117ζ11ζ116ζ115116111011411911311811211711115    linear of order 22 faithful
ρ1311ζ11ζ117ζ112ζ118ζ113ζ119ζ114ζ1110ζ115ζ116ζ115ζ11ζ117ζ112ζ118ζ113ζ119ζ114ζ1110ζ116    linear of order 11
ρ141-1ζ11ζ117ζ112ζ118ζ113ζ119ζ114ζ1110ζ115ζ116115111171121181131191141110116    linear of order 22 faithful
ρ1511ζ113ζ1110ζ116ζ112ζ119ζ115ζ11ζ118ζ114ζ117ζ114ζ113ζ1110ζ116ζ112ζ119ζ115ζ11ζ118ζ117    linear of order 11
ρ161-1ζ113ζ1110ζ116ζ112ζ119ζ115ζ11ζ118ζ114ζ117114113111011611211911511118117    linear of order 22 faithful
ρ1711ζ115ζ112ζ1110ζ117ζ114ζ11ζ119ζ116ζ113ζ118ζ113ζ115ζ112ζ1110ζ117ζ114ζ11ζ119ζ116ζ118    linear of order 11
ρ181-1ζ115ζ112ζ1110ζ117ζ114ζ11ζ119ζ116ζ113ζ118113115112111011711411119116118    linear of order 22 faithful
ρ1911ζ117ζ115ζ113ζ11ζ1110ζ118ζ116ζ114ζ112ζ119ζ112ζ117ζ115ζ113ζ11ζ1110ζ118ζ116ζ114ζ119    linear of order 11
ρ201-1ζ117ζ115ζ113ζ11ζ1110ζ118ζ116ζ114ζ112ζ119112117115113111110118116114119    linear of order 22 faithful
ρ2111ζ119ζ118ζ117ζ116ζ115ζ114ζ113ζ112ζ11ζ1110ζ11ζ119ζ118ζ117ζ116ζ115ζ114ζ113ζ112ζ1110    linear of order 11
ρ221-1ζ119ζ118ζ117ζ116ζ115ζ114ζ113ζ112ζ11ζ1110111191181171161151141131121110    linear of order 22 faithful

Permutation representations of C22
Regular action on 22 points - transitive group 22T1
Generators in S22
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22)

G:=sub<Sym(22)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)])

G:=TransitiveGroup(22,1);

Polynomial with Galois group C22 over ℚ
actionf(x)Disc(f)
22T1x22+x21+x20+x19+x18+x17+x16+x15+x14+x13+x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1-2321

Matrix representation of C22 in GL1(𝔽23) generated by

15
G:=sub<GL(1,GF(23))| [15] >;

C22 in GAP, Magma, Sage, TeX

C_{22}
% in TeX

G:=Group("C22");
// GroupNames label

G:=SmallGroup(22,2);
// by ID

G=gap.SmallGroup(22,2);
# by ID

G:=PCGroup([2,-2,-11]);
// Polycyclic

G:=Group<a|a^22=1>;
// generators/relations

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