Properties

Label 22T24
Order \(12100\)
n \(22\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $22$
Transitive number $t$ :  $24$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,10,11,3)(2,9,8,5,7)(12,18,15,22,13)(14,17,21,19,20), (1,22)(2,14,9,13,10,16,7,18,5,12,11,19,4,20,3,17,6,15,8,21)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
5:  $C_5$
10:  $D_{5}$, $C_{10}$
20:  20T1, 20T2
50:  $D_5\times C_5$
100:  20T25

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: None

Low degree siblings

44T138

Siblings are shown 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 $ $1$ $1$ $()$
$ 11, 11 $ $100$ $11$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,13,14,15,16,17,18,19,20,21,22)$
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $11$ $(12,14,16,18,20,22,13,15,17,19,21)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,17,15,16,21)(14,22,18,20,19)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,15,21,17,16)(14,18,19,22,20)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,21,16,15,17)(14,19,20,18,22)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,16,17,21,15)(14,20,22,19,18)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $5$ $(13,21,16,15,17)(14,19,20,18,22)$
$ 11, 5, 5, 1 $ $220$ $55$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,13,22,15,18)(14,20,19,21,17)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)$
$ 11, 5, 5, 1 $ $220$ $55$ $( 1,10, 4, 8, 9)( 3, 5,11, 7, 6)(12,13,14,15,16,17,18,19,20,21,22)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,17,15,16,21)(14,22,18,20,19)$
$ 5, 5, 5, 5, 1, 1 $ $121$ $5$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,16,17,21,15)(14,20,22,19,18)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $5$ $(13,16,17,21,15)(14,20,22,19,18)$
$ 11, 5, 5, 1 $ $220$ $55$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,13,17,22,20)(14,21,16,18,15)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,15,21,17,16)(14,18,19,22,20)$
$ 5, 5, 5, 5, 1, 1 $ $121$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,17,15,16,21)(14,22,18,20,19)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $5$ $(13,17,15,16,21)(14,22,18,20,19)$
$ 11, 5, 5, 1 $ $220$ $55$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,13,18,21,14)(15,17,16,22,19)$
$ 5, 5, 5, 5, 1, 1 $ $121$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,15,21,17,16)(14,18,19,22,20)$
$ 5, 5, 5, 5, 1, 1 $ $121$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,21,16,15,17)(14,19,20,18,22)$
$ 10, 10, 1, 1 $ $121$ $10$ $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,20,21,18,16,22,15,14,17,19)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)(13,19,17,14,15,22,16,18,21,20)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(13,14,16,20,17,22,21,19,15,18)$
$ 10, 2, 2, 2, 2, 2, 1, 1 $ $242$ $10$ $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,18,15,19,21,22,17,20,16,14)$
$ 10, 2, 2, 2, 2, 2, 1, 1 $ $242$ $10$ $( 2, 8, 6, 3, 4,11, 5, 7,10, 9)(13,22)(14,21)(15,20)(16,19)(17,18)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,18,15,19,21,22,17,20,16,14)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)(13,20,21,18,16,22,15,14,17,19)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(13,19,17,14,15,22,16,18,21,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $121$ $2$ $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,22)(14,21)(15,20)(16,19)(17,18)$
$ 10, 2, 2, 2, 2, 2, 1, 1 $ $242$ $10$ $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,22)(14,21)(15,20)(16,19)(17,18)$
$ 10, 2, 2, 2, 2, 2, 1, 1 $ $242$ $10$ $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,14,16,20,17,22,21,19,15,18)$
$ 10, 10, 1, 1 $ $121$ $10$ $( 2, 8, 6, 3, 4,11, 5, 7,10, 9)(13,19,17,14,15,22,16,18,21,20)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,19,17,14,15,22,16,18,21,20)$
$ 10, 10, 1, 1 $ $121$ $10$ $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)(13,14,16,20,17,22,21,19,15,18)$
$ 10, 10, 1, 1 $ $121$ $10$ $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(13,18,15,19,21,22,17,20,16,14)$
$ 20, 2 $ $605$ $20$ $( 1,22)( 2,14, 9,13,10,16, 7,18, 5,12,11,19, 4,20, 3,17, 6,15, 8,21)$
$ 4, 4, 4, 4, 4, 2 $ $605$ $4$ $( 1,18, 2,22)( 3,15,11,14)( 4,19,10,21)( 5,12, 9,17)( 6,16, 8,13)( 7,20)$
$ 20, 2 $ $605$ $20$ $( 1,20, 8,17, 2,18, 4,14, 7,19, 6,21,10,13, 5,12, 3,16,11,22)( 9,15)$
$ 20, 2 $ $605$ $20$ $( 1,14,11,20, 9,21, 5,12, 8,16, 3,13, 4,18, 6,17,10,15, 7,22)( 2,19)$
$ 20, 2 $ $605$ $20$ $( 1,19, 5,12, 7,14, 8,15, 3,21, 6,13, 2,20,11,18,10,17, 4,22)( 9,16)$
$ 20, 2 $ $605$ $20$ $( 1,15, 2,17, 8,18,11,13, 7,16, 5,12, 4,21, 9,20, 6,14,10,22)( 3,19)$
$ 20, 2 $ $605$ $20$ $( 1,16, 2,15, 4,13, 8,20, 5,12,10,18, 9,19, 7,21, 3,14, 6,22)(11,17)$
$ 20, 2 $ $605$ $20$ $( 1,21, 7,13,11,15,10,20, 2,16, 4,17, 9,14, 5,12, 6,18, 3,22)( 8,19)$
$ 20, 2 $ $605$ $20$ $( 1,17, 7,15, 5,12, 2,13, 3,20,10,14, 4,16, 6,19, 9,18, 8,22)(11,21)$
$ 4, 4, 4, 4, 4, 2 $ $605$ $4$ $( 1,13, 9,22)( 2,21, 8,14)( 3,18, 7,17)( 4,15, 6,20)( 5,12)(10,19,11,16)$

Group invariants

Order:  $12100=2^{2} \cdot 5^{2} \cdot 11^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.