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G = Dic11  order 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

Derived series
C1 — C11 — Dic11
Chief series
C1 — C11 — C22 — Dic11
Lower central
C11 — Dic11
Upper central
C1 — C2

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

C1
C2
C11
11C4
C22
Dic11

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
ρ52200ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ1110+ζ11ζ1110+ζ11ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ118+ζ113    orthogonal lifted from D11
ρ62200ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ118+ζ113ζ118+ζ113ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ119+ζ112    orthogonal lifted from D11
ρ72200ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ116+ζ115ζ116+ζ115ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ117+ζ114    orthogonal lifted from D11
ρ82200ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ117+ζ114ζ117+ζ114ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ1110+ζ11    orthogonal lifted from D11
ρ92200ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ119+ζ112ζ119+ζ112ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ116+ζ115    orthogonal lifted from D11
ρ102-200ζ119+ζ112ζ118+ζ113ζ116+ζ115ζ1110+ζ11ζ117+ζ114-ζ117-ζ114-ζ119-ζ112-ζ118-ζ113-ζ116-ζ115-ζ1110-ζ11    symplectic faithful, Schur index 2
ρ112-200ζ116+ζ115ζ119+ζ112ζ117+ζ114ζ118+ζ113ζ1110+ζ11-ζ1110-ζ11-ζ116-ζ115-ζ119-ζ112-ζ117-ζ114-ζ118-ζ113    symplectic faithful, Schur index 2
ρ122-200ζ1110+ζ11ζ117+ζ114ζ118+ζ113ζ116+ζ115ζ119+ζ112-ζ119-ζ112-ζ1110-ζ11-ζ117-ζ114-ζ118-ζ113-ζ116-ζ115    symplectic faithful, Schur index 2
ρ132-200ζ117+ζ114ζ116+ζ115ζ1110+ζ11ζ119+ζ112ζ118+ζ113-ζ118-ζ113-ζ117-ζ114-ζ116-ζ115-ζ1110-ζ11-ζ119-ζ112    symplectic faithful, Schur index 2
ρ142-200ζ118+ζ113ζ1110+ζ11ζ119+ζ112ζ117+ζ114ζ116+ζ115-ζ116-ζ115-ζ118-ζ113-ζ1110-ζ11-ζ119-ζ112-ζ117-ζ114    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 24 12 35)(2 23 13 34)(3 44 14 33)(4 43 15 32)(5 42 16 31)(6 41 17 30)(7 40 18 29)(8 39 19 28)(9 38 20 27)(10 37 21 26)(11 36 22 25)
 
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), (1,24,12,35)(2,23,13,34)(3,44,14,33)(4,43,15,32)(5,42,16,31)(6,41,17,30)(7,40,18,29)(8,39,19,28)(9,38,20,27)(10,37,21,26)(11,36,22,25)>;
 
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), (1,24,12,35)(2,23,13,34)(3,44,14,33)(4,43,15,32)(5,42,16,31)(6,41,17,30)(7,40,18,29)(8,39,19,28)(9,38,20,27)(10,37,21,26)(11,36,22,25) );
 
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)], [(1,24,12,35),(2,23,13,34),(3,44,14,33),(4,43,15,32),(5,42,16,31),(6,41,17,30),(7,40,18,29),(8,39,19,28),(9,38,20,27),(10,37,21,26),(11,36,22,25)]])
 

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
 

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

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