Properties

 Label 12T16 Order $$36$$ n $$12$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $S_3^2$

Related objects

Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $16$ Group : $S_3^2$ CHM label : $[3^{2}]E(4)$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (2,10,6)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) $|\Aut(F/K)|$: $6$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3^2$

Low degree siblings

6T9, 9T8, 18T9, 18T11 x 2, 36T13

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 1, 1, 1, 1, 1, 1$ $4$ $3$ $( 2, 6,10)( 4,12, 8)$ $2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 2)( 3, 8)( 4,11)( 5, 6)( 7,12)( 9,10)$ $6, 6$ $6$ $6$ $( 1, 2, 5, 6, 9,10)( 3, 8,11, 4, 7,12)$ $2, 2, 2, 2, 2, 2$ $9$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$ $6, 6$ $6$ $6$ $( 1, 4, 9,12, 5, 8)( 2, 3, 6, 7,10,11)$ $2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ $3, 3, 3, 3$ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3,11, 7)( 4,12, 8)$ $3, 3, 3, 3$ $2$ $3$ $( 1, 5, 9)( 2,10, 6)( 3,11, 7)( 4, 8,12)$

Group invariants

 Order: $36=2^{2} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [36, 10]
 Character table:  2 2 . 2 1 2 1 2 1 1 3 2 2 1 1 . 1 1 2 2 1a 3a 2a 6a 2b 6b 2c 3b 3c 2P 1a 3a 1a 3b 1a 3c 1a 3b 3c 3P 1a 1a 2a 2a 2b 2c 2c 1a 1a 5P 1a 3a 2a 6a 2b 6b 2c 3b 3c X.1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 -1 1 1 1 1 X.3 1 1 -1 -1 1 -1 -1 1 1 X.4 1 1 1 1 -1 -1 -1 1 1 X.5 2 -1 . . . -1 2 2 -1 X.6 2 -1 . . . 1 -2 2 -1 X.7 2 -1 -2 1 . . . -1 2 X.8 2 -1 2 -1 . . . -1 2 X.9 4 1 . . . . . -2 -2