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Properties

Label 22T25
Degree $22$
Order $12100$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{11}^2:(C_5\times D_{10})$

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Show commands: Magma

magma: G := TransitiveGroup(22, 25);
 

Group action invariants

Degree $n$:  $22$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $25$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{11}^2:(C_5\times D_{10})$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8,3,5,2)(4,9,7,10,11)(12,22,13,20,17)(14,18,21,15,16), (1,10)(2,9)(3,8)(4,7)(5,6)(12,14,19,15,16,13,22,17,21,20), (1,22,6,18,10,17,11,14,3,16)(2,19,9,20,8,12,5,21,7,15)(4,13)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5$:  $C_5$
$10$:  $D_{5}$, $C_{10}$ x 3
$20$:  $D_{10}$, 20T3
$50$:  $D_5\times C_5$
$100$:  20T24

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: None

Low degree siblings

44T139

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 $ $50$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,14,16,18,20,22,13,15,17,19,21)$
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $11$ $(12,19,15,22,18,14,21,17,13,20,16)$
$ 11, 11 $ $50$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,21,19,17,15,13,22,20,18,16,14)$
$ 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, 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, 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 $ $220$ $55$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,14,13,19,16)(15,18,22,20,21)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $5$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)$
$ 11, 5, 5, 1 $ $220$ $55$ $( 1, 6, 4, 7, 8)( 2,10, 9, 5,11)(12,14,16,18,20,22,13,15,17,19,21)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,21,16,15,17)(14,19,20,18,22)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $5$ $(13,15,21,17,16)(14,18,19,22,20)$
$ 11, 5, 5, 1 $ $220$ $55$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,14,20,16,15)(13,17,18,21,19)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,16,17,21,15)(14,20,22,19,18)$
$ 5, 5, 5, 5, 1, 1 $ $121$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,21,16,15,17)(14,19,20,18,22)$
$ 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 $ $220$ $55$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,14,21,18,13)(15,19,22,16,17)$
$ 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, 5, 5, 1, 1 $ $121$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,17,15,16,21)(14,22,18,20,19)$
$ 10, 2, 2, 2, 2, 2, 1, 1 $ $242$ $10$ $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,20,21,18,16,22,15,14,17,19)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 8, 6, 3, 4,11, 5, 7,10, 9)(13,18,15,19,21,22,17,20,16,14)$
$ 10, 2, 2, 2, 2, 2, 1, 1 $ $242$ $10$ $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(13,22)(14,21)(15,20)(16,19)(17,18)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,19,17,14,15,22,16,18,21,20)$
$ 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, 2, 2, 2, 2, 2, 1, 1 $ $242$ $10$ $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,19,17,14,15,22,16,18,21,20)$
$ 10, 10, 1, 1 $ $121$ $10$ $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(13,18,15,19,21,22,17,20,16,14)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,14,16,20,17,22,21,19,15,18)$
$ 10, 2, 2, 2, 2, 2, 1, 1 $ $242$ $10$ $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)(13,22)(14,21)(15,20)(16,19)(17,18)$
$ 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 $ $242$ $10$ $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(13,20,21,18,16,22,15,14,17,19)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)(13,18,15,19,21,22,17,20,16,14)$
$ 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 $ $242$ $10$ $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)(13,19,17,14,15,22,16,18,21,20)$
$ 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, 10, 2 $ $605$ $10$ $( 1,22, 6,18,10,17,11,14, 3,16)( 2,19, 9,20, 8,12, 5,21, 7,15)( 4,13)$
$ 10, 10, 2 $ $605$ $10$ $( 1,14, 9,18, 4,21, 3,15, 5,16)( 2,20, 7,17, 8,12, 6,22,10,13)(11,19)$
$ 10, 10, 2 $ $605$ $10$ $( 1,19, 8,12,10,21, 9,22, 4,16)( 2,18)( 3,17, 7,13, 5,15, 6,14,11,20)$
$ 22 $ $550$ $22$ $( 1,18, 6,20,11,22, 5,13,10,15, 4,17, 9,19, 3,21, 8,12, 2,14, 7,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $55$ $2$ $( 1,17)( 2,13)( 3,20)( 4,16)( 5,12)( 6,19)( 7,15)( 8,22)( 9,18)(10,14)(11,21)$
$ 10, 10, 2 $ $605$ $10$ $( 1,20, 9,14, 8,12, 4,15,10,16)( 2,22)( 3,13, 6,19, 7,21,11,18, 5,17)$
$ 10, 10, 2 $ $605$ $10$ $( 1,15,11,17, 2,13, 9,21, 6,16)( 3,22, 7,14,10,19, 4,20, 5,18)( 8,12)$
$ 10, 10, 2 $ $605$ $10$ $( 1,17, 5,22, 3,14, 4,18, 9,16)( 2,21,10,20, 6,15, 8,12, 7,19)(11,13)$
$ 10, 10, 2 $ $605$ $10$ $( 1,13, 8,12, 3,19, 5,14, 2,16)( 4,22, 9,15, 7,20,10,18,11,21)( 6,17)$
$ 10, 10, 2 $ $605$ $10$ $( 1,16)( 2,17, 4,19,10,14, 6,21, 5,20)( 3,18, 7,22, 8,12,11,15, 9,13)$
$ 22 $ $550$ $22$ $( 1,21, 2,15, 3,20, 4,14, 5,19, 6,13, 7,18, 8,12, 9,17,10,22,11,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $55$ $2$ $( 1,19)( 2,13)( 3,18)( 4,12)( 5,17)( 6,22)( 7,16)( 8,21)( 9,15)(10,20)(11,14)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $12100=2^{2} \cdot 5^{2} \cdot 11^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  12100.o
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);