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

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.