# Properties

 Label 18T10 Order $$36$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_3^2 : C_4$

# Related objects

## Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $10$ Group : $C_3^2 : C_4$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,13,10,15)(2,14,9,16)(3,5,7,18)(4,6,8,17)(11,12), (1,14)(2,13)(3,12)(4,11)(5,9)(6,10)(15,18)(16,17) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$ x 2

Degree 9: $C_3^2:C_4$

## Low degree siblings

6T10 x 2, 9T9, 12T17 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $9$ $2$ $( 3,17)( 4,18)( 5,15)( 6,16)( 7,14)( 8,13)( 9,11)(10,12)$ $4, 4, 4, 4, 2$ $9$ $4$ $( 1, 2)( 3, 9,17,11)( 4,10,18,12)( 5, 7,15,14)( 6, 8,16,13)$ $4, 4, 4, 4, 2$ $9$ $4$ $( 1, 2)( 3,11,17, 9)( 4,12,18,10)( 5,14,15, 7)( 6,13,16, 8)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1, 3,17)( 2, 4,18)( 5, 8, 9)( 6, 7,10)(11,13,15)(12,14,16)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1, 6,16)( 2, 5,15)( 3, 7,12)( 4, 8,11)( 9,13,18)(10,14,17)$

## Group invariants

 Order: $36=2^{2} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [36, 9]
 Character table:  2 2 2 2 2 . . 3 2 . . . 2 2 1a 2a 4a 4b 3a 3b 2P 1a 1a 2a 2a 3a 3b 3P 1a 2a 4b 4a 1a 1a X.1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 X.3 1 -1 A -A 1 1 X.4 1 -1 -A A 1 1 X.5 4 . . . 1 -2 X.6 4 . . . -2 1 A = -E(4) = -Sqrt(-1) = -i