Properties

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

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

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.