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Properties

Label 12T300
Order \(239500800\)
n \(12\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $A_{12}$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $300$
Group :  $A_{12}$
CHM label :  $A12$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3), (1,2)(3,4,5,6,7,8,9,10,11,12)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 1, 1, 1 $ $492800$ $3$ $( 3, 8, 6)( 4, 5,11)( 9,12,10)$
$ 9, 1, 1, 1 $ $8870400$ $9$ $( 3,11,10, 8, 4, 9, 6, 5,12)$
$ 3, 3, 3, 3 $ $246400$ $3$ $( 1, 2, 7)( 3, 6, 8)( 4,11, 5)( 9,10,12)$
$ 9, 3 $ $8870400$ $9$ $( 1,12,11, 2, 9, 5, 7,10, 4)( 3, 6, 8)$
$ 9, 3 $ $8870400$ $9$ $( 1,11, 9, 7, 4,12, 2, 5,10)( 3, 8, 6)$
$ 2, 2, 2, 2, 2, 2 $ $10395$ $2$ $( 1, 2)( 3, 6)( 4,12)( 5, 8)( 7,11)( 9,10)$
$ 5, 5, 1, 1 $ $4790016$ $5$ $( 1, 5, 7, 6,12)( 2, 8,11, 3, 4)$
$ 10, 2 $ $23950080$ $10$ $( 1, 3, 5, 4, 7, 2, 6, 8,12,11)( 9,10)$
$ 5, 1, 1, 1, 1, 1, 1, 1 $ $19008$ $5$ $( 1, 7,12, 5, 6)$
$ 7, 1, 1, 1, 1, 1 $ $570240$ $7$ $( 2, 8,10,11, 4, 9, 3)$
$ 7, 5 $ $6842880$ $35$ $( 1, 5,12, 7, 6)( 2, 9,11, 8, 3, 4,10)$
$ 7, 5 $ $6842880$ $35$ $( 1,12, 6, 5, 7)( 2, 9,11, 8, 3, 4,10)$
$ 11, 1 $ $21772800$ $11$ $( 1, 4, 7,10, 5, 8, 2,12, 6, 3, 9)$
$ 11, 1 $ $21772800$ $11$ $( 1, 9, 3, 6,12, 2, 8, 5,10, 7, 4)$
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $440$ $3$ $( 3, 6, 8)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $1485$ $2$ $( 4, 7)( 9,12)$
$ 4, 2, 1, 1, 1, 1, 1, 1 $ $83160$ $4$ $( 1, 2)( 4,12, 7, 9)$
$ 3, 2, 2, 1, 1, 1, 1, 1 $ $166320$ $6$ $( 4, 7)( 5,11,10)( 9,12)$
$ 4, 3, 2, 1, 1, 1 $ $3326400$ $12$ $( 1, 2)( 4, 9, 7,12)( 5,10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $51975$ $2$ $( 1, 2)( 4, 7)( 6, 8)( 9,12)$
$ 3, 2, 2, 2, 2, 1 $ $415800$ $6$ $( 1, 2)( 4, 7)( 5,11,10)( 6, 8)( 9,12)$
$ 6, 3, 2, 1 $ $13305600$ $6$ $( 1, 2)( 4, 6, 9, 7, 8,12)( 5,10,11)$
$ 7, 3, 1, 1 $ $11404800$ $21$ $( 1, 2,11,12, 9, 4,10)( 3, 6, 8)$
$ 7, 2, 2, 1 $ $8553600$ $14$ $( 1,12)( 2,10, 4, 3, 8,11, 9)( 5, 7)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $36960$ $3$ $( 1,11,10)( 2, 5, 8)$
$ 3, 3, 2, 2, 1, 1 $ $1663200$ $6$ $( 1,10,11)( 2, 8, 5)( 4, 7)( 9,12)$
$ 4, 3, 3, 2 $ $3326400$ $12$ $( 1,11,10)( 2, 5, 8)( 3, 6)( 4,12, 7, 9)$
$ 4, 2, 2, 2, 1, 1 $ $1247400$ $4$ $( 1, 2)( 4, 9, 7,12)( 6,11)( 8,10)$
$ 4, 4, 1, 1, 1, 1 $ $623700$ $4$ $( 1,11,10, 3)( 4,12, 7, 9)$
$ 4, 4, 3, 1 $ $4989600$ $12$ $( 1, 3,10,11)( 2, 6, 8)( 4, 9, 7,12)$
$ 5, 2, 2, 1, 1, 1 $ $1995840$ $10$ $( 2,10)( 3, 6)( 4,11, 5, 7,12)$
$ 5, 4, 2, 1 $ $11975040$ $20$ $( 1, 9)( 2, 3,10, 6)( 4, 7,11,12, 5)$
$ 5, 3, 1, 1, 1, 1 $ $1330560$ $15$ $( 2, 4,11)( 3,12, 9, 6, 5)$
$ 5, 3, 2, 2 $ $3991680$ $30$ $( 1, 8)( 2,11, 4)( 3, 6,12, 5, 9)( 7,10)$
$ 4, 4, 2, 2 $ $1871100$ $4$ $( 1, 2)( 3, 8, 5,11)( 4, 9, 7,12)( 6,10)$
$ 6, 2, 2, 2 $ $1663200$ $6$ $( 1, 2,11, 5,10, 8)( 3, 6)( 4, 7)( 9,12)$
$ 6, 2, 1, 1, 1, 1 $ $1663200$ $6$ $( 1, 8,10, 5,11, 2)( 9,12)$
$ 6, 4, 1, 1 $ $9979200$ $12$ $( 1, 5,11, 8,10, 2)( 4, 9, 7,12)$
$ 8, 4 $ $14968800$ $8$ $( 1, 3,11, 5,10, 8, 6, 2)( 4,12, 7, 9)$
$ 5, 3, 3, 1 $ $5322240$ $15$ $( 1, 9,12, 4, 5)( 3, 8, 6)( 7,11,10)$
$ 6, 6 $ $6652800$ $6$ $( 1, 2,10, 8,11, 5)( 3, 7, 9, 4,12, 6)$
$ 8, 2, 1, 1 $ $14968800$ $8$ $( 3,12, 8, 4, 5, 9,11, 7)( 6,10)$

Group invariants

Order:  $239500800=2^{9} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.