Properties

 Label 12T58 Order $$96$$ n $$12$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_2^2\wr C_2:C_3$

Related objects

Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $58$ Group : $C_2^2\wr C_2:C_3$ CHM label : $[2^{4}]6$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,3,5,7,9,11)(2,4,6,8,10,12), (1,12)(8,9), (1,12)(4,5) $|\Aut(F/K)|$: $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$

Low degree siblings

8T33 x 2, 12T58, 12T59 x 2, 16T183, 24T181 x 2, 24T182 x 2, 24T183 x 2, 24T184 x 2, 24T185, 24T186, 32T389

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $6$ $2$ $( 6, 7)(10,11)$ $2, 2, 2, 2, 1, 1, 1, 1$ $6$ $2$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$ $2, 2, 2, 2, 1, 1, 1, 1$ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$ $6, 6$ $16$ $6$ $( 1, 2, 4, 6, 8,11)( 3, 5, 7, 9,10,12)$ $3, 3, 3, 3$ $16$ $3$ $( 1, 4, 8)( 2, 6,10)( 3, 7,11)( 5, 9,12)$ $4, 4, 2, 2$ $12$ $4$ $( 1, 6)( 2, 8, 3, 9)( 4,10, 5,11)( 7,12)$ $2, 2, 2, 2, 2, 2$ $4$ $2$ $( 1, 6)( 2, 8)( 3, 9)( 4,11)( 5,10)( 7,12)$ $3, 3, 3, 3$ $16$ $3$ $( 1, 8, 4)( 2,10, 6)( 3,11, 7)( 5,12, 9)$ $6, 6$ $16$ $6$ $( 1,10, 9, 6, 4, 2)( 3,12,11, 8, 7, 5)$

Group invariants

 Order: $96=2^{5} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [96, 70]
 Character table: 2 5 4 4 5 1 1 3 3 1 1 3 1 . . . 1 1 . 1 1 1 1a 2a 2b 2c 6a 3a 4a 2d 3b 6b 2P 1a 1a 1a 1a 3a 3b 2c 1a 3a 3b 3P 1a 2a 2b 2c 2d 1a 4a 2d 1a 2d 5P 1a 2a 2b 2c 6b 3b 4a 2d 3a 6a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 1 -1 -1 1 -1 X.3 1 1 1 1 A -/A -1 -1 -A /A X.4 1 1 1 1 /A -A -1 -1 -/A A X.5 1 1 1 1 -/A -A 1 1 -/A -A X.6 1 1 1 1 -A -/A 1 1 -A -/A X.7 3 -1 -1 3 . . -1 3 . . X.8 3 -1 -1 3 . . 1 -3 . . X.9 6 -2 2 -2 . . . . . . X.10 6 2 -2 -2 . . . . . . A = -E(3) = (1-Sqrt(-3))/2 = -b3