# Properties

 Label 12T272 Order $$7920$$ n $$12$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $M_{11}$

# Related objects

## Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $272$ Group : $M_{11}$ CHM label : $M_{11}(12)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,7,3,10,5,9,6,12)(2,11,8,4), (1,6,3,9)(2,7,12,10,4,5,11,8) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

## Low degree siblings

11T6, 22T22

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 1, 1$ $1584$ $5$ $( 1,12,10, 2, 9)( 3, 4, 7, 8, 5)$ $11, 1$ $720$ $11$ $( 1,12,11, 8, 7, 9, 4, 6, 2, 3,10)$ $11, 1$ $720$ $11$ $( 1,10, 3, 2, 6, 4, 9, 7, 8,11,12)$ $2, 2, 2, 2, 1, 1, 1, 1$ $165$ $2$ $( 1, 2)( 3,12)( 5, 9)( 8,10)$ $4, 4, 2, 2$ $990$ $4$ $( 1, 3, 2,12)( 4,11)( 5,10, 9, 8)( 6, 7)$ $8, 4$ $990$ $8$ $( 1,10,12, 5, 2, 8, 3, 9)( 4, 7,11, 6)$ $8, 4$ $990$ $8$ $( 1, 9, 3, 8, 2, 5,12,10)( 4, 6,11, 7)$ $3, 3, 3, 1, 1, 1$ $440$ $3$ $( 1, 4, 2)( 3,11,12)( 5, 9, 7)$ $6, 3, 2, 1$ $1320$ $6$ $( 1, 2, 4)( 3, 5,11, 9,12, 7)( 8,10)$

## Group invariants

 Order: $7920=2^{4} \cdot 3^{2} \cdot 5 \cdot 11$ Cyclic: No Abelian: No Solvable: No GAP id: Data not available
 Character table:  2 4 . . . 4 1 1 3 3 3 3 2 . . . 1 2 1 . . . 5 1 . . 1 . . . . . . 11 1 1 1 . . . . . . . 1a 11a 11b 5a 2a 3a 6a 4a 8a 8b 2P 1a 11b 11a 5a 1a 3a 3a 2a 4a 4a 3P 1a 11a 11b 5a 2a 1a 2a 4a 8a 8b 5P 1a 11a 11b 1a 2a 3a 6a 4a 8b 8a 7P 1a 11b 11a 5a 2a 3a 6a 4a 8b 8a 11P 1a 1a 1a 5a 2a 3a 6a 4a 8a 8b X.1 1 1 1 1 1 1 1 1 1 1 X.2 10 -1 -1 . 2 1 -1 2 . . X.3 10 -1 -1 . -2 1 1 . B -B X.4 10 -1 -1 . -2 1 1 . -B B X.5 11 . . 1 3 2 . -1 -1 -1 X.6 16 A /A 1 . -2 . . . . X.7 16 /A A 1 . -2 . . . . X.8 44 . . -1 4 -1 1 . . . X.9 45 1 1 . -3 . . 1 -1 -1 X.10 55 . . . -1 1 -1 -1 1 1 A = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10 = (-1-Sqrt(-11))/2 = -1-b11 B = -E(8)-E(8)^3 = -Sqrt(-2) = -i2