Copied to
clipboard

G = D33order 66 = 2·3·11

Dihedral group

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

Aliases: D33, C11⋊S3, C3⋊D11, C331C2, sometimes denoted D66 or Dih33 or Dih66, SmallGroup(66,3)

Series: Derived Chief Lower central Upper central

C1C33 — D33
C1C11C33 — D33
C33 — D33
C1

Generators and relations for D33
 G = < a,b | a33=b2=1, bab=a-1 >

33C2
11S3
3D11

Character table of D33

 class 12311A11B11C11D11E33A33B33C33D33E33F33G33H33I33J
 size 1332222222222222222
ρ1111111111111111111    trivial
ρ21-11111111111111111    linear of order 2
ρ320-122222-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ4202ζ119112ζ117114ζ116115ζ118113ζ111011ζ119112ζ118113ζ117114ζ116115ζ116115ζ117114ζ118113ζ119112ζ111011ζ111011    orthogonal lifted from D11
ρ5202ζ118113ζ116115ζ119112ζ111011ζ117114ζ118113ζ111011ζ116115ζ119112ζ119112ζ116115ζ111011ζ118113ζ117114ζ117114    orthogonal lifted from D11
ρ6202ζ111011ζ119112ζ118113ζ117114ζ116115ζ111011ζ117114ζ119112ζ118113ζ118113ζ119112ζ117114ζ111011ζ116115ζ116115    orthogonal lifted from D11
ρ7202ζ116115ζ111011ζ117114ζ119112ζ118113ζ116115ζ119112ζ111011ζ117114ζ117114ζ111011ζ119112ζ116115ζ118113ζ118113    orthogonal lifted from D11
ρ8202ζ117114ζ118113ζ111011ζ116115ζ119112ζ117114ζ116115ζ118113ζ111011ζ111011ζ118113ζ116115ζ117114ζ119112ζ119112    orthogonal lifted from D11
ρ920-1ζ119112ζ117114ζ116115ζ118113ζ1110113ζ1193ζ1121193ζ1183ζ1131183ζ1173ζ114117ζ32ζ11632ζ11511532ζ11632ζ11511632ζ11732ζ114117ζ3ζ1183ζ113113ζ3ζ1193ζ112112ζ3ζ11103ζ1111ζ32ζ111032ζ1111    orthogonal faithful
ρ1020-1ζ118113ζ116115ζ119112ζ111011ζ117114ζ3ζ1183ζ113113ζ32ζ111032ζ1111ζ32ζ11632ζ115115ζ3ζ1193ζ1121123ζ1193ζ11211932ζ11632ζ115116ζ3ζ11103ζ11113ζ1183ζ11311832ζ11732ζ1141173ζ1173ζ114117    orthogonal faithful
ρ1120-1ζ111011ζ119112ζ118113ζ117114ζ116115ζ32ζ111032ζ111132ζ11732ζ1141173ζ1193ζ112119ζ3ζ1183ζ1131133ζ1183ζ113118ζ3ζ1193ζ1121123ζ1173ζ114117ζ3ζ11103ζ1111ζ32ζ11632ζ11511532ζ11632ζ115116    orthogonal faithful
ρ1220-1ζ117114ζ118113ζ111011ζ116115ζ11911232ζ11732ζ114117ζ32ζ11632ζ1151153ζ1183ζ113118ζ32ζ111032ζ1111ζ3ζ11103ζ1111ζ3ζ1183ζ11311332ζ11632ζ1151163ζ1173ζ1141173ζ1193ζ112119ζ3ζ1193ζ112112    orthogonal faithful
ρ1320-1ζ119112ζ117114ζ116115ζ118113ζ111011ζ3ζ1193ζ112112ζ3ζ1183ζ11311332ζ11732ζ11411732ζ11632ζ115116ζ32ζ11632ζ1151153ζ1173ζ1141173ζ1183ζ1131183ζ1193ζ112119ζ32ζ111032ζ1111ζ3ζ11103ζ1111    orthogonal faithful
ρ1420-1ζ117114ζ118113ζ111011ζ116115ζ1191123ζ1173ζ11411732ζ11632ζ115116ζ3ζ1183ζ113113ζ3ζ11103ζ1111ζ32ζ111032ζ11113ζ1183ζ113118ζ32ζ11632ζ11511532ζ11732ζ114117ζ3ζ1193ζ1121123ζ1193ζ112119    orthogonal faithful
ρ1520-1ζ111011ζ119112ζ118113ζ117114ζ116115ζ3ζ11103ζ11113ζ1173ζ114117ζ3ζ1193ζ1121123ζ1183ζ113118ζ3ζ1183ζ1131133ζ1193ζ11211932ζ11732ζ114117ζ32ζ111032ζ111132ζ11632ζ115116ζ32ζ11632ζ115115    orthogonal faithful
ρ1620-1ζ116115ζ111011ζ117114ζ119112ζ118113ζ32ζ11632ζ1151153ζ1193ζ112119ζ3ζ11103ζ111132ζ11732ζ1141173ζ1173ζ114117ζ32ζ111032ζ1111ζ3ζ1193ζ11211232ζ11632ζ1151163ζ1183ζ113118ζ3ζ1183ζ113113    orthogonal faithful
ρ1720-1ζ118113ζ116115ζ119112ζ111011ζ1171143ζ1183ζ113118ζ3ζ11103ζ111132ζ11632ζ1151163ζ1193ζ112119ζ3ζ1193ζ112112ζ32ζ11632ζ115115ζ32ζ111032ζ1111ζ3ζ1183ζ1131133ζ1173ζ11411732ζ11732ζ114117    orthogonal faithful
ρ1820-1ζ116115ζ111011ζ117114ζ119112ζ11811332ζ11632ζ115116ζ3ζ1193ζ112112ζ32ζ111032ζ11113ζ1173ζ11411732ζ11732ζ114117ζ3ζ11103ζ11113ζ1193ζ112119ζ32ζ11632ζ115115ζ3ζ1183ζ1131133ζ1183ζ113118    orthogonal faithful

Smallest permutation representation of D33
On 33 points
Generators in S33
(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)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)

G:=sub<Sym(33)| (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)>;

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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18) );

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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18)])

D33 is a maximal subgroup of   S3×D11  D99  C3⋊D33  C11⋊S4  C3⋊F11  D165  D231
D33 is a maximal quotient of   Dic33  D99  C3⋊D33  C11⋊S4  D165  D231

Matrix representation of D33 in GL2(𝔽67) generated by

3242
3358
,
115
066
G:=sub<GL(2,GF(67))| [32,33,42,58],[1,0,15,66] >;

D33 in GAP, Magma, Sage, TeX

D_{33}
% in TeX

G:=Group("D33");
// GroupNames label

G:=SmallGroup(66,3);
// by ID

G=gap.SmallGroup(66,3);
# by ID

G:=PCGroup([3,-2,-3,-11,25,542]);
// Polycyclic

G:=Group<a,b|a^33=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D33 in TeX
Character table of D33 in TeX

׿
×
𝔽