Properties

Label 26T32
Order \(4056\)
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 $t$ :  $32$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,23,12,26,4,25,11,21,13,18,8,19)(2,15,3,20,7,14,10,16,9,24,5,17)(6,22), (1,23,5,18,4,16)(2,25,8,24,13,21)(3,14,11,17,9,26)(6,20,7,22,10,15)(12,19)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
8:  $D_{4}$
12:  $C_6\times C_2$
24:  $D_4 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

26T32, 26T37

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, 1, 1 $ $1$ $1$ $()$
$ 13, 13 $ $12$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 13 $ $12$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $(14,23,19,15,24,20,16,25,21,17,26,22,18)$
$ 13, 13 $ $24$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$
$ 13, 13 $ $24$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 13, 13 $ $24$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$
$ 13, 13 $ $24$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $(14,19,24,16,21,26,18,23,15,20,25,17,22)$
$ 13, 13 $ $24$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $169$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
$ 6, 6, 6, 6, 1, 1 $ $169$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 6, 6, 6, 6, 1, 1 $ $169$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$
$ 12, 12, 2 $ $338$ $12$ $( 1,23,12,26, 4,25,11,21,13,18, 8,19)( 2,15, 3,20, 7,14,10,16, 9,24, 5,17) ( 6,22)$
$ 12, 12, 2 $ $338$ $12$ $( 1,15, 7,14, 2,17, 4,21,11,22, 3,19)( 5,23, 8,16,12,24,13,26,10,20, 6,25) ( 9,18)$
$ 4, 4, 4, 4, 4, 4, 2 $ $338$ $4$ $( 1,17,10,19)( 2,23, 9,26)( 3,16, 8,20)( 4,22, 7,14)( 5,15, 6,21)(11,25,13,24) (12,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $2$ $(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $156$ $26$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,22)(15,21)(16,20)(17,19)(23,26) (24,25)$
$ 13, 2, 2, 2, 2, 2, 2, 1 $ $156$ $26$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17)(15,16)(18,26)(19,25)(20,24) (21,23)$
$ 6, 6, 3, 3, 3, 3, 1, 1 $ $338$ $6$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,24,23,26,17,18)(16,21,19,25,20,22)$
$ 6, 6, 3, 3, 3, 3, 1, 1 $ $338$ $6$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 6, 6, 6, 6, 2 $ $338$ $6$ $( 1,23, 8,19, 6,22)( 2,15, 4,25, 9,24)( 3,20,13,18,12,26)( 5,17) ( 7,14,10,16,11,21)$
$ 6, 6, 6, 6, 2 $ $338$ $6$ $( 1,15,10,20,11,22)( 2,17,13,26, 7,14)( 3,19)( 4,21, 6,25,12,24) ( 5,23, 9,18, 8,16)$
$ 26 $ $156$ $26$ $( 1,17,11,25, 8,20, 5,15, 2,23,12,18, 9,26, 6,21, 3,16,13,24,10,19, 7,14, 4,22 )$
$ 26 $ $156$ $26$ $( 1,20,12,21,10,22, 8,23, 6,24, 4,25, 2,26,13,14,11,15, 9,16, 7,17, 5,18, 3,19 )$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $26$ $2$ $( 1,26)( 2,19)( 3,25)( 4,18)( 5,24)( 6,17)( 7,23)( 8,16)( 9,22)(10,15)(11,21) (12,14)(13,20)$

Group invariants

Order:  $4056=2^{3} \cdot 3 \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.