Properties

 Label 18T35 Order $$72$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $PSU(3,2)$

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Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $35$ Group : $PSU(3,2)$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (3,15,17,5)(4,16,18,6)(7,11,13,9)(8,12,14,10), (1,7,12,6)(2,8,11,5)(3,18,15,13)(4,17,16,14)(9,10) $|\Aut(F/K)|$: $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $C_3^2:Q_8$

Low degree siblings

9T14, 12T47, 18T35 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 4, 4, 1, 1$ $18$ $4$ $( 3, 5,17,15)( 4, 6,18,16)( 7, 9,13,11)( 8,10,14,12)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $9$ $2$ $( 3,17)( 4,18)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11)(10,12)$ $4, 4, 4, 4, 2$ $18$ $4$ $( 1, 2)( 3, 7,17,13)( 4, 8,18,14)( 5,11,15, 9)( 6,12,16,10)$ $4, 4, 4, 4, 2$ $18$ $4$ $( 1, 2)( 3, 9,17,11)( 4,10,18,12)( 5, 7,15,13)( 6, 8,16,14)$ $3, 3, 3, 3, 3, 3$ $8$ $3$ $( 1, 3,17)( 2, 4,18)( 5, 8,10)( 6, 7, 9)(11,13,16)(12,14,15)$

Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [72, 41]
 Character table:  2 3 2 3 2 2 . 3 2 . . . . 2 1a 4a 2a 4b 4c 3a 2P 1a 2a 1a 2a 2a 3a 3P 1a 4a 2a 4b 4c 1a X.1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 1 X.3 1 -1 1 1 -1 1 X.4 1 1 1 -1 -1 1 X.5 2 . -2 . . 2 X.6 8 . . . . -1