Properties

Label 26T45
Order \(8112\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Learn more about

Group action invariants

Degree $n$ :  $26$
Transitive number $t$ :  $45$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,13,22,11,15,7,14,12,25,9,21,3,26,4,23,6,17,10,18,5,20,8,24)(2,16), (1,21,3,25,7,20,2,23,5,16,11,15,10,26,8,22,4,14,9,24,6,18,13,19)(12,17)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_4$ x 2, $C_2^2$
6:  $C_6$ x 3
8:  $C_4\times C_2$
12:  $C_{12}$ x 2, $C_6\times C_2$
16:  $C_8:C_2$
24:  24T2
48:  24T16

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

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.