# Properties

 Label 10T31 Order $$720$$ n $$10$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $M_{10}$

# Related objects

## Group action invariants

 Degree $n$ : $10$ Transitive number $t$ : $31$ Group : $M_{10}$ CHM label : $M(10)=L(10)'2$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,2)(4,7)(5,8)(9,10), (1,4,2,8)(3,7,6,5), (1,2,10)(3,4,5)(6,7,8), (1,3,2,6)(4,5,8,7) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 5: None

## Low degree siblings

12T181, 20T148, 20T150 x 2, 30T162, 36T1253, 40T591, 45T109

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 1, 1$ $180$ $4$ $( 3, 4, 6, 5)( 7,10, 8, 9)$ $2, 2, 2, 2, 1, 1$ $45$ $2$ $( 3, 6)( 4, 5)( 7, 8)( 9,10)$ $4, 4, 1, 1$ $90$ $4$ $( 3, 9, 6,10)( 4, 8, 5, 7)$ $3, 3, 3, 1$ $80$ $3$ $( 2, 3, 6)( 4, 9, 7)( 5, 8,10)$ $8, 2$ $90$ $8$ $( 1, 2)( 3, 4,10, 7, 6, 5, 9, 8)$ $8, 2$ $90$ $8$ $( 1, 2)( 3, 5,10, 8, 6, 4, 9, 7)$ $5, 5$ $144$ $5$ $( 1, 2, 3, 4, 9)( 5, 7,10, 6, 8)$

## Group invariants

 Order: $720=2^{4} \cdot 3^{2} \cdot 5$ Cyclic: No Abelian: No Solvable: No GAP id: [720, 765]
 Character table:  2 4 2 4 3 . 3 3 . 3 2 . . . 2 . . . 5 1 . . . . . . 1 1a 4a 2a 4b 3a 8a 8b 5a 2P 1a 2a 1a 2a 3a 4b 4b 5a 3P 1a 4a 2a 4b 1a 8a 8b 5a 5P 1a 4a 2a 4b 3a 8b 8a 1a 7P 1a 4a 2a 4b 3a 8b 8a 5a X.1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 1 -1 -1 1 X.3 9 -1 1 1 . 1 1 -1 X.4 9 1 1 1 . -1 -1 -1 X.5 10 . 2 -2 1 . . . X.6 10 . -2 . 1 A -A . X.7 10 . -2 . 1 -A A . X.8 16 . . . -2 . . 1 A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2