Properties

Label 26T50
Order \(12168\)
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 $t$ :  $50$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,23,7,25,5,20,10,26,4,24,6,16)(2,19,11,22,8,21,9,17,13,14,3,15)(12,18), (1,19,4,25,8,20,9,22,6,16,2,21)(3,23,11,26,13,17,7,18,12,15,10,24)(5,14)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
8:  $Q_8$
12:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
24:  24T4, 24T5
36:  $C_6\times S_3$
72:  24T64

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $24$ $13$ $(14,26,25,24,23,22,21,20,19,18,17,16,15)$
$ 13, 13 $ $72$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 13, 13 $ $72$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$
$ 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 $ $169$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $3$ $(15,17,23)(16,20,19)(18,26,24)(21,22,25)$
$ 13, 3, 3, 3, 3, 1 $ $312$ $39$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $338$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $26$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)$
$ 13, 3, 3, 3, 3, 1 $ $312$ $39$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$
$ 6, 6, 2, 2, 2, 2, 2, 2, 1, 1 $ $338$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$
$ 6, 6, 2, 2, 2, 2, 2, 2, 1, 1 $ $338$ $6$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,24,23,26,17,18)(16,21,19,25,20, 22)$
$ 6, 6, 6, 6, 1, 1 $ $338$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$
$ 12, 12, 2 $ $1014$ $12$ $( 1,23, 7,25, 5,20,10,26, 4,24, 6,16)( 2,19,11,22, 8,21, 9,17,13,14, 3,15) (12,18)$
$ 12, 12, 2 $ $1014$ $12$ $( 1,15, 4,18, 8,22, 9,23, 6,20, 2,16)( 3,17,11,25,13,14, 7,21,12,26,10,24) ( 5,19)$
$ 4, 4, 4, 4, 4, 4, 2 $ $1014$ $4$ $( 1,17, 5,16)( 2,20, 4,26)( 3,23)( 6,19,13,14)( 7,22,12,24)( 8,25,11,21) ( 9,15,10,18)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,20,24,22,23,16,26,21,17,19,18,25)$
$ 12, 4, 4, 4, 1, 1 $ $338$ $12$ $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,19,26,22)(16,24,25,17)(18,21,23,20)$
$ 12, 4, 4, 4, 1, 1 $ $338$ $12$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,16,18,22,17,20,26,25,23,19,24,21)$
$ 12, 4, 4, 4, 1, 1 $ $338$ $12$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,25,18,19,17,21,26,16,23,22,24,20)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,21,24,19,23,25,26,20,17,22,18,16)$
$ 12, 4, 4, 4, 1, 1 $ $338$ $12$ $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,22,26,19)(16,17,25,24)(18,20,23,21)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,16,18,22,17,20,26,25,23,19,24,21)$
$ 12, 12, 1, 1 $ $338$ $12$ $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,20,24,22,23,16,26,21,17,19,18,25)$
$ 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, 2 $ $1014$ $12$ $( 1,23)( 2,18, 5,16, 4,21,13,15,10,17,11,25)( 3,26, 9,22, 7,19,12,20, 6,24, 8, 14)$
$ 12, 12, 2 $ $1014$ $12$ $( 1,15, 2,26,12,19, 8,14, 7,16,10,23)( 3,24, 9,25, 4,22, 6,18,13,17, 5,20) (11,21)$
$ 4, 4, 4, 4, 4, 4, 2 $ $1014$ $4$ $( 1,17,13,23)( 2,24,12,16)( 3,18,11,22)( 4,25,10,15)( 5,19, 9,21)( 6,26, 8,14) ( 7,20)$

Group invariants

Order:  $12168=2^{3} \cdot 3^{2} \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.