# Properties

 Label 23T5 Order $$10200960$$ n $$23$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $M_{23}$

## Group action invariants

 Degree $n$ : $23$ Transitive number $t$ : $5$ Group : $M_{23}$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15), (2,14,18,20,8)(3,7,12,13,19)(4,21,17,15,10)(5,11,16,6,9), (3,19)(4,14)(5,20)(6,10)(8,15)(11,18)(17,21)(22,23), (1,22)(2,10)(3,14)(4,17)(8,15)(9,11)(13,20)(19,21) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - 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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1$ $56672$ $3$ $( 1, 6, 8)( 4,22, 9)( 5,14,10)(13,20,21)(15,23,17)(16,19,18)$ $5, 5, 5, 5, 1, 1, 1$ $680064$ $5$ $( 1, 4,14,17,20)( 2, 7,11, 3,12)( 5,23,13, 8, 9)( 6,22,10,15,21)$ $15, 5, 3$ $680064$ $15$ $( 1, 5,21, 4,23, 6,14,13,22,17, 8,10,20, 9,15)( 2,11,12, 7, 3)(16,18,19)$ $15, 5, 3$ $680064$ $15$ $( 1,10,13, 4,15, 8,14,21, 9,17, 6, 5,20,22,23)( 2,11,12, 7, 3)(16,19,18)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1$ $3795$ $2$ $( 1,11)( 2,18)( 3,21)( 6,22)( 7,15)( 8,13)(10,17)(19,23)$ $6, 6, 3, 3, 2, 2, 1$ $850080$ $6$ $( 1, 6,23,11,22,19)( 2, 3,17,18,21,10)( 4, 9,14)( 5,20,12)( 7,15)( 8,13)$ $4, 4, 4, 4, 2, 2, 1, 1, 1$ $318780$ $4$ $( 1, 8,11,22)( 2,17,12,16)( 3,19,14, 6)( 4,10)( 7,21)( 9,13,15,23)$ $8, 8, 4, 2, 1$ $1275120$ $8$ $( 1, 9, 8,13,11,15,22,23)( 2, 6,17, 3,12,19,16,14)( 4,21,10, 7)( 5,20)$ $11, 11, 1$ $927360$ $11$ $( 1,22,20, 4,11,21,13, 8, 9, 6, 3)( 2,19,17,16, 5,12,10,18,14,15,23)$ $11, 11, 1$ $927360$ $11$ $( 1, 3, 6, 9, 8,13,21,11, 4,20,22)( 2,23,15,14,18,10,12, 5,16,17,19)$ $7, 7, 7, 1, 1$ $728640$ $7$ $( 1, 8,14, 6,17,21,22)( 2,19,16,12,11,18, 7)( 4, 9,13,10, 5,23,15)$ $7, 7, 7, 1, 1$ $728640$ $7$ $( 1,22,21,17, 6,14, 8)( 2, 7,18,11,12,16,19)( 4,15,23, 5,10,13, 9)$ $14, 7, 2$ $728640$ $14$ $( 1,14,17,22, 8, 6,21)( 2,10,11, 4,19, 5,18, 9,16,23, 7,13,12,15)( 3,20)$ $14, 7, 2$ $728640$ $14$ $( 1,22,21,17, 6,14, 8)( 2, 4,18,23,12,10,19, 9, 7,15,11, 5,16,13)( 3,20)$ $23$ $443520$ $23$ $( 1, 2,15,13,11,20, 9,18,12,21, 7, 4, 8,16,19, 5,10,23,14,17, 6,22, 3)$ $23$ $443520$ $23$ $( 1, 3,22, 6,17,14,23,10, 5,19,16, 8, 4, 7,21,12,18, 9,20,11,13,15, 2)$

## Group invariants

 Order: $10200960=2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 23$ Cyclic: No Abelian: No Solvable: No GAP id: Data not available
 Character table: Data not available.