Properties

Label 23T5
Degree $23$
Order $10200960$
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)
$|\Aut(F/K)|$:  $1$
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)

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

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:  not available
Character table: not available.