metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D36, C4⋊D9, C9⋊1D4, C3.D12, C36⋊1C2, C6.8D6, D18⋊1C2, C12.2S3, C2.4D18, C18.3C22, sometimes denoted D72 or Dih36 or Dih72, SmallGroup(72,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D36
G = < a,b | a36=b2=1, bab=a-1 >
Character table of D36
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | |
| size | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | -1 | -1 | -1 | -2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ12 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ13 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | √3 | -√3 | -√3 | -√3 | √3 | √3 | orthogonal lifted from D12 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | -√3 | √3 | √3 | √3 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | √3 | -√3 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ4ζ95-ζ4ζ94 | ζ4ζ97-ζ4ζ92 | ζ43ζ98-ζ43ζ9 | -ζ4ζ95+ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | orthogonal faithful |
| ρ17 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -√3 | √3 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ4ζ97-ζ4ζ92 | -ζ43ζ98+ζ43ζ9 | ζ4ζ95-ζ4ζ94 | -ζ4ζ97+ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | ζ43ζ98-ζ43ζ9 | orthogonal faithful |
| ρ18 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | √3 | -√3 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ4ζ97+ζ4ζ92 | ζ43ζ98-ζ43ζ9 | -ζ4ζ95+ζ4ζ94 | ζ4ζ97-ζ4ζ92 | ζ4ζ95-ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | orthogonal faithful |
| ρ19 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | √3 | -√3 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ43ζ98+ζ43ζ9 | -ζ4ζ95+ζ4ζ94 | ζ4ζ97-ζ4ζ92 | ζ43ζ98-ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ95-ζ4ζ94 | orthogonal faithful |
| ρ20 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -√3 | √3 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ43ζ98-ζ43ζ9 | ζ4ζ95-ζ4ζ94 | -ζ4ζ97+ζ4ζ92 | -ζ43ζ98+ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | orthogonal faithful |
| ρ21 | 2 | -2 | 0 | 0 | -1 | 0 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -√3 | √3 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ4ζ95+ζ4ζ94 | -ζ4ζ97+ζ4ζ92 | -ζ43ζ98+ζ43ζ9 | ζ4ζ95-ζ4ζ94 | ζ43ζ98-ζ43ζ9 | ζ4ζ97-ζ4ζ92 | orthogonal faithful |
►On 36 points
Generators in S36
(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)
(1 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)
G:=sub<Sym(36)| (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;
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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );
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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)]])
D36 is a maximal subgroup of
C72⋊C2 D72 D4⋊D9 Q8⋊2D9 D36⋊5C2 D4×D9 Q8⋊3D9 D108 C3⋊D36 D36⋊C3 C36⋊S3 C22⋊D36 C12.4S4 C5⋊D36 D180
D36 is a maximal quotient of
Dic36 C72⋊C2 D72 C4⋊Dic9 D18⋊C4 D108 C3⋊D36 C36⋊S3 C22⋊D36 C5⋊D36 D180
Matrix representation of D36 ►in GL2(𝔽37) generated by
| 12 | 4 |
| 33 | 8 |
| 12 | 4 |
| 29 | 25 |
G:=sub<GL(2,GF(37))| [12,33,4,8],[12,29,4,25] >;
D36 in GAP, Magma, Sage, TeX
D_{36} % in TeX
G:=Group("D36"); // GroupNames label
G:=SmallGroup(72,6);
// by ID
G=gap.SmallGroup(72,6);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,61,26,803,138,1204]);
// Polycyclic
G:=Group<a,b|a^36=b^2=1,b*a*b=a^-1>;
// generators/relations
Export
Subgroup lattice of D36 in TeX
Character table of D36 in TeX