Properties

Label 26T44
Order \(8112\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $44$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,11,5,10,8)(2,6,7,4,13,12)(14,15,24)(16,20,17)(18,25,23)(19,21,26), (1,17,3,24,13,20,11,26)(2,14,8,22,12,23,6,15)(4,21,5,18,10,16,9,19)(7,25)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$, $C_3 : C_4$ x 2
16:  $C_8:C_2$
24:  24T6
48:  24T20

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: 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, 1 $ $1$ $1$ $()$
$ 13, 13 $ $48$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$
$ 13, 13 $ $48$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$
$ 13, 13 $ $48$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $24$ $13$ $(14,20,26,19,25,18,24,17,23,16,22,15,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $338$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
$ 6, 6, 3, 3, 3, 3, 1, 1 $ $338$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,23,17)(16,19,20)(18,24,26)(21,25,22)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $312$ $26$ $( 1,13)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(14,21,15,22,16,23,17,24,18,25,19, 26,20)$
$ 6, 6, 3, 3, 3, 3, 1, 1 $ $338$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,17,23)(16,20,19)(18,26,24)(21,22,25)$
$ 6, 6, 6, 6, 1, 1 $ $338$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $169$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,16,18,22,17,20,26,25,23,19,24,21)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,20,24,22,23,16,26,21,17,19,18,25)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $338$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,16,18,22,17,20,26,25,23,19,24,21)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,21,24,19,23,25,26,20,17,22,18,16)$
$ 4, 4, 4, 4, 4, 4, 1, 1 $ $169$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
$ 8, 8, 8, 2 $ $1014$ $8$ $( 1,17, 3,24,13,20,11,26)( 2,14, 8,22,12,23, 6,15)( 4,21, 5,18,10,16, 9,19) ( 7,25)$
$ 8, 8, 8, 2 $ $1014$ $8$ $( 1,15,11,18,13,16, 3,26)( 2,14, 6,23,12,17, 8,21)( 4,25, 9,20,10,19, 5,24) ( 7,22)$
$ 8, 8, 8, 2 $ $1014$ $8$ $( 1,16, 2,14,10,24, 9,26)( 3,25, 5,21, 8,15, 6,19)( 4,23,13,18, 7,17,11,22) (12,20)$
$ 8, 8, 8, 2 $ $1014$ $8$ $( 1,19, 8,23, 4,17,10,26)( 2,14,13,24, 3,22, 5,25)( 6,20, 7,15,12,16,11,21) ( 9,18)$

Group invariants

Order:  $8112=2^{4} \cdot 3 \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.