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G = Dic11order 44 = 22·11

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic11, C11⋊C4, C22.C2, C2.D11, SmallGroup(44,1)

Series: Derived Chief Lower central Upper central

C1C11 — Dic11
C1C11C22 — Dic11
C11 — Dic11
C1C2

Generators and relations for Dic11
 G = < a,b | a22=1, b2=a11, bab-1=a-1 >

11C4

Character table of Dic11

 class 124A4B11A11B11C11D11E22A22B22C22D22E
 size 1111112222222222
ρ111111111111111    trivial
ρ211-1-11111111111    linear of order 2
ρ31-1-ii11111-1-1-1-1-1    linear of order 4
ρ41-1i-i11111-1-1-1-1-1    linear of order 4
ρ52200ζ116115ζ119112ζ117114ζ118113ζ111011ζ111011ζ116115ζ119112ζ117114ζ118113    orthogonal lifted from D11
ρ62200ζ117114ζ116115ζ111011ζ119112ζ118113ζ118113ζ117114ζ116115ζ111011ζ119112    orthogonal lifted from D11
ρ72200ζ118113ζ111011ζ119112ζ117114ζ116115ζ116115ζ118113ζ111011ζ119112ζ117114    orthogonal lifted from D11
ρ82200ζ119112ζ118113ζ116115ζ111011ζ117114ζ117114ζ119112ζ118113ζ116115ζ111011    orthogonal lifted from D11
ρ92200ζ111011ζ117114ζ118113ζ116115ζ119112ζ119112ζ111011ζ117114ζ118113ζ116115    orthogonal lifted from D11
ρ102-200ζ119112ζ118113ζ116115ζ111011ζ117114117114119112118113116115111011    symplectic faithful, Schur index 2
ρ112-200ζ116115ζ119112ζ117114ζ118113ζ111011111011116115119112117114118113    symplectic faithful, Schur index 2
ρ122-200ζ111011ζ117114ζ118113ζ116115ζ119112119112111011117114118113116115    symplectic faithful, Schur index 2
ρ132-200ζ117114ζ116115ζ111011ζ119112ζ118113118113117114116115111011119112    symplectic faithful, Schur index 2
ρ142-200ζ118113ζ111011ζ119112ζ117114ζ116115116115118113111011119112117114    symplectic faithful, Schur index 2

Smallest permutation representation of Dic11
Regular action on 44 points
Generators in S44
(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 41 42 43 44)
(1 37 12 26)(2 36 13 25)(3 35 14 24)(4 34 15 23)(5 33 16 44)(6 32 17 43)(7 31 18 42)(8 30 19 41)(9 29 20 40)(10 28 21 39)(11 27 22 38)

G:=sub<Sym(44)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44), (1,37,12,26)(2,36,13,25)(3,35,14,24)(4,34,15,23)(5,33,16,44)(6,32,17,43)(7,31,18,42)(8,30,19,41)(9,29,20,40)(10,28,21,39)(11,27,22,38)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44), (1,37,12,26)(2,36,13,25)(3,35,14,24)(4,34,15,23)(5,33,16,44)(6,32,17,43)(7,31,18,42)(8,30,19,41)(9,29,20,40)(10,28,21,39)(11,27,22,38) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22),(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)], [(1,37,12,26),(2,36,13,25),(3,35,14,24),(4,34,15,23),(5,33,16,44),(6,32,17,43),(7,31,18,42),(8,30,19,41),(9,29,20,40),(10,28,21,39),(11,27,22,38)])

Dic11 is a maximal subgroup of
C4×D11  C11⋊D4  C11⋊C20  C11⋊F5  C32⋊Dic11  C11⋊Dic11
 Dic11p: Dic22  Dic33  Dic55  Dic77  Dic121 ...
Dic11 is a maximal quotient of
C11⋊F5  C32⋊Dic11
 C2p.D11: C11⋊C8  Dic33  Dic55  Dic77  Dic121  C11⋊Dic11 ...

Matrix representation of Dic11 in GL2(𝔽23) generated by

225
520
,
022
10
G:=sub<GL(2,GF(23))| [22,5,5,20],[0,1,22,0] >;

Dic11 in GAP, Magma, Sage, TeX

{\rm Dic}_{11}
% in TeX

G:=Group("Dic11");
// GroupNames label

G:=SmallGroup(44,1);
// by ID

G=gap.SmallGroup(44,1);
# by ID

G:=PCGroup([3,-2,-2,-11,6,362]);
// Polycyclic

G:=Group<a,b|a^22=1,b^2=a^11,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic11 in TeX
Character table of Dic11 in TeX

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