Properties

 Label 12T26 Order $$48$$ n $$12$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_2^2 \times A_4$

Related objects

Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $26$ Group : $C_2^2 \times A_4$ CHM label : $A_{4}(12)x2^{2}$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,4)(2,5)(3,9)(6,12)(7,10)(8,11), (1,7)(2,11)(3,12)(4,10)(5,8)(6,9), (1,11,6)(2,9,7)(3,10,5)(4,8,12) $|\Aut(F/K)|$: $4$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4\times C_2$ x 3

Low degree siblings

12T25 x 3, 12T26, 16T58, 24T49 x 3, 24T50

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1, 1, 1$ $3$ $2$ $( 2, 5)( 3, 6)( 8,11)( 9,12)$ $2, 2, 2, 2, 1, 1, 1, 1$ $3$ $2$ $( 2, 8)( 3,12)( 5,11)( 6, 9)$ $2, 2, 2, 2, 1, 1, 1, 1$ $3$ $2$ $( 2,11)( 3, 9)( 5, 8)( 6,12)$ $6, 6$ $4$ $6$ $( 1, 2, 3,10, 8, 6)( 4, 5, 9, 7,11,12)$ $6, 6$ $4$ $6$ $( 1, 2, 6, 7,11, 9)( 3, 4, 5,12,10, 8)$ $6, 6$ $4$ $6$ $( 1, 2, 9, 4, 5, 3)( 6,10, 8,12, 7,11)$ $3, 3, 3, 3$ $4$ $3$ $( 1, 2,12)( 3, 7,11)( 4, 5, 6)( 8, 9,10)$ $6, 6$ $4$ $6$ $( 1, 3, 5, 4, 9, 2)( 6,11, 7,12, 8,10)$ $6, 6$ $4$ $6$ $( 1, 3,11,10, 6, 5)( 2, 4, 9, 8, 7,12)$ $6, 6$ $4$ $6$ $( 1, 3, 2, 7,12,11)( 4, 9, 5,10, 6, 8)$ $3, 3, 3, 3$ $4$ $3$ $( 1, 3, 8)( 2,10, 6)( 4, 9,11)( 5, 7,12)$ $2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 4)( 2, 5)( 3, 9)( 6,12)( 7,10)( 8,11)$ $2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ $2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 7)( 2,11)( 3,12)( 4,10)( 5, 8)( 6, 9)$ $2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1,10)( 2, 8)( 3, 6)( 4, 7)( 5,11)( 9,12)$

Group invariants

 Order: $48=2^{4} \cdot 3$ Cyclic: No Abelian: No Solvable: Yes GAP id: [48, 49]
 Character table:  2 4 4 4 4 2 2 2 2 2 2 2 2 4 4 4 4 3 1 . . . 1 1 1 1 1 1 1 1 1 . 1 1 1a 2a 2b 2c 6a 6b 6c 3a 6d 6e 6f 3b 2d 2e 2f 2g 2P 1a 1a 1a 1a 3b 3b 3b 3b 3a 3a 3a 3a 1a 1a 1a 1a 3P 1a 2a 2b 2c 2g 2f 2d 1a 2d 2g 2f 1a 2d 2e 2f 2g 5P 1a 2a 2b 2c 6e 6f 6d 3b 6c 6a 6b 3a 2d 2e 2f 2g X.1 1 1 1 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 -1 1 -1 1 X.3 1 -1 1 -1 -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 -1 1 1 -1 X.5 1 -1 -1 1 A -A -A A -/A /A -/A /A -1 1 -1 1 X.6 1 -1 -1 1 /A -/A -/A /A -A A -A A -1 1 -1 1 X.7 1 -1 1 -1 -/A -/A /A /A A -A -A A 1 1 -1 -1 X.8 1 -1 1 -1 -A -A A A /A -/A -/A /A 1 1 -1 -1 X.9 1 1 -1 -1 -/A /A -/A /A -A -A A A -1 1 1 -1 X.10 1 1 -1 -1 -A A -A A -/A -/A /A /A -1 1 1 -1 X.11 1 1 1 1 A A A A /A /A /A /A 1 1 1 1 X.12 1 1 1 1 /A /A /A /A A A A A 1 1 1 1 X.13 3 -1 -1 -1 . . . . . . . . 3 -1 3 3 X.14 3 -1 1 1 . . . . . . . . -3 -1 3 -3 X.15 3 1 -1 1 . . . . . . . . 3 -1 -3 -3 X.16 3 1 1 -1 . . . . . . . . -3 -1 -3 3 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3