# Properties

 Label 18T38 Order $$72$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_2^2:D_9$

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## Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $38$ Group : $C_2^2:D_9$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,15)(2,16)(3,14)(4,13)(5,18)(6,17)(9,12)(10,11), (1,8,2,7)(3,12,4,11)(5,9,6,10)(13,14)(15,18)(16,17) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
18:  $D_{9}$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4$

Degree 9: $D_{9}$

## Low degree siblings

18T39

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

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [72, 15]
 Character table:  2 3 3 2 2 2 2 . . . 3 2 1 . . 1 2 2 2 2 1a 2a 2b 4a 6a 3a 9a 9b 9c 2P 1a 1a 1a 2a 3a 3a 9b 9c 9a 3P 1a 2a 2b 4a 2a 1a 3a 3a 3a 5P 1a 2a 2b 4a 6a 3a 9c 9a 9b 7P 1a 2a 2b 4a 6a 3a 9b 9c 9a 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 A C B X.5 2 2 . . -1 -1 B A C X.6 2 2 . . -1 -1 C B A X.7 3 -1 -1 1 -1 3 . . . X.8 3 -1 1 -1 -1 3 . . . X.9 6 -2 . . 1 -3 . . . A = E(9)^4+E(9)^5 B = E(9)^2+E(9)^7 C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7