# Properties

 Label 18T13 Order $$36$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_{18}$

# Related objects

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
18:  $D_{9}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 9: $D_{9}$

## Low degree siblings

18T13

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

## Group invariants

 Order: $36=2^{2} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [36, 4]
 Character table:  2 2 2 2 2 1 1 1 1 1 1 1 1 3 2 . 2 . 2 2 2 2 2 2 2 2 1a 2a 2b 2c 9a 18a 9b 18b 6a 3a 18c 9c 2P 1a 1a 1a 1a 9b 9b 9c 9c 3a 3a 9a 9a 3P 1a 2a 2b 2c 3a 6a 3a 6a 2b 1a 6a 3a 5P 1a 2a 2b 2c 9c 18c 9a 18a 6a 3a 18b 9b 7P 1a 2a 2b 2c 9b 18b 9c 18c 6a 3a 18a 9a 11P 1a 2a 2b 2c 9b 18b 9c 18c 6a 3a 18a 9a 13P 1a 2a 2b 2c 9c 18c 9a 18a 6a 3a 18b 9b 17P 1a 2a 2b 2c 9a 18a 9b 18b 6a 3a 18c 9c X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 X.3 1 -1 1 -1 1 1 1 1 1 1 1 1 X.4 1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 X.5 2 . 2 . -1 -1 -1 -1 2 2 -1 -1 X.6 2 . -2 . -1 1 -1 1 -2 2 1 -1 X.7 2 . 2 . A A B B -1 -1 C C X.8 2 . 2 . B B C C -1 -1 A A X.9 2 . 2 . C C A A -1 -1 B B X.10 2 . -2 . A -A B -B 1 -1 -C C X.11 2 . -2 . B -B C -C 1 -1 -A A X.12 2 . -2 . C -C A -A 1 -1 -B B A = E(9)^2+E(9)^7 B = E(9)^4+E(9)^5 C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7