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## G = Dic11order 44 = 22·11

### Dicyclic group

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 >

Character table of Dic11

 class 1 2 4A 4B 11A 11B 11C 11D 11E 22A 22B 22C 22D 22E size 1 1 11 11 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 -i i 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 4 ρ4 1 -1 i -i 1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 4 ρ5 2 2 0 0 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 orthogonal lifted from D11 ρ6 2 2 0 0 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 orthogonal lifted from D11 ρ7 2 2 0 0 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 orthogonal lifted from D11 ρ8 2 2 0 0 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 orthogonal lifted from D11 ρ9 2 2 0 0 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 orthogonal lifted from D11 ρ10 2 -2 0 0 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 -ζ117-ζ114 -ζ119-ζ112 -ζ118-ζ113 -ζ116-ζ115 -ζ1110-ζ11 symplectic faithful, Schur index 2 ρ11 2 -2 0 0 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 -ζ1110-ζ11 -ζ116-ζ115 -ζ119-ζ112 -ζ117-ζ114 -ζ118-ζ113 symplectic faithful, Schur index 2 ρ12 2 -2 0 0 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 -ζ119-ζ112 -ζ1110-ζ11 -ζ117-ζ114 -ζ118-ζ113 -ζ116-ζ115 symplectic faithful, Schur index 2 ρ13 2 -2 0 0 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 -ζ118-ζ113 -ζ117-ζ114 -ζ116-ζ115 -ζ1110-ζ11 -ζ119-ζ112 symplectic faithful, Schur index 2 ρ14 2 -2 0 0 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+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

 22 5 5 20
,
 0 22 1 0
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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