Properties

 Label 18T37 Order $$72$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_3:S_4$

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Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $37$ Group : $C_3:S_4$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,6,3,2,5,4)(7,12,10,8,11,9)(13,18,16)(14,17,15), (1,6)(2,5)(3,4)(7,16,8,15)(9,14,10,13)(11,18,12,17), (1,16)(2,15)(3,13)(4,14)(5,18)(6,17)(7,11)(8,12) $|\Aut(F/K)|$: $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $C_3^2:C_2$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$ x 4

Degree 6: $S_4$

Degree 9: $C_3^2:C_2$

Low degree siblings

12T44 x 3, 18T40

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

Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [72, 43]
 Character table:  2 3 3 2 2 2 2 . . . 3 2 1 . . 1 2 2 2 2 1a 2a 2b 4a 6a 3a 3b 3c 3d 2P 1a 1a 1a 2a 3a 3a 3b 3c 3d 3P 1a 2a 2b 4a 2a 1a 1a 1a 1a 5P 1a 2a 2b 4a 6a 3a 3b 3c 3d X.1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 1 1 1 X.3 2 2 . . 2 2 -1 -1 -1 X.4 2 2 . . -1 -1 2 -1 -1 X.5 2 2 . . -1 -1 -1 -1 2 X.6 2 2 . . -1 -1 -1 2 -1 X.7 3 -1 -1 1 -1 3 . . . X.8 3 -1 1 -1 -1 3 . . . X.9 6 -2 . . 1 -3 . . .