Properties

 Label 18T12 Order $$36$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_2\times C_3:S_3$

Related objects

Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $12$ Group : $C_2\times C_3:S_3$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,17)(2,18)(3,4)(5,13)(6,14)(7,12)(8,11)(9,16)(10,15), (1,15)(2,16)(3,13)(4,14)(5,6)(7,9)(8,10)(11,17)(12,18), (1,11)(2,12)(3,15)(4,16)(5,8)(6,7)(13,18)(14,17) $|\Aut(F/K)|$: $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 4
12:  $D_{6}$ x 4
18:  $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 6: $D_{6}$ x 4

Degree 9: $C_3^2:C_2$

Low degree siblings

18T12

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

Group invariants

 Order: $36=2^{2} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [36, 13]
 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 6a 3a 6b 3b 3c 6c 6d 3d 2P 1a 1a 1a 1a 3a 3a 3b 3b 3c 3c 3d 3d 3P 1a 2a 2b 2c 2b 1a 2b 1a 1a 2b 2b 1a 5P 1a 2a 2b 2c 6a 3a 6b 3b 3c 6c 6d 3d 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 . 2 2 -1 -1 -1 -1 -1 -1 X.6 2 . -2 . -2 2 1 -1 -1 1 1 -1 X.7 2 . 2 . -1 -1 2 2 -1 -1 -1 -1 X.8 2 . -2 . 1 -1 -2 2 -1 1 1 -1 X.9 2 . -2 . 1 -1 1 -1 -1 1 -2 2 X.10 2 . -2 . 1 -1 1 -1 2 -2 1 -1 X.11 2 . 2 . -1 -1 -1 -1 -1 -1 2 2 X.12 2 . 2 . -1 -1 -1 -1 2 2 -1 -1