Properties

Label 28T26
Order \(168\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times F_7$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $26$
Group :  $C_4\times F_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,21,26,18,5)(2,10,22,25,17,6)(3,19,16,24,7,11)(4,20,15,23,8,12), (1,16,2,15)(3,6,23,18,19,10,4,5,24,17,20,9)(7,14,12,21,27,25,8,13,11,22,28,26)
$|\Aut(F/K)|$:  $4$

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, $C_6\times C_2$
24:  24T2
42:  $F_7$
84:  $F_7 \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 7: $F_7$

Degree 14: $F_7 \times C_2$

Low degree siblings

28T26

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 $ $1$ $1$ $()$
$ 6, 6, 6, 6, 1, 1, 1, 1 $ $7$ $6$ $( 3, 7,19,27,24,11)( 4, 8,20,28,23,12)( 5,13, 9,26,18,21)( 6,14,10,25,17,22)$
$ 6, 6, 6, 6, 1, 1, 1, 1 $ $7$ $6$ $( 3,11,24,27,19, 7)( 4,12,23,28,20, 8)( 5,21,18,26, 9,13)( 6,22,17,25,10,14)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3,19,24)( 4,20,23)( 5, 9,18)( 6,10,17)( 7,27,11)( 8,28,12)(13,26,21) (14,25,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3,24,19)( 4,23,20)( 5,18, 9)( 6,17,10)( 7,11,27)( 8,12,28)(13,21,26) (14,22,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $7$ $2$ $( 3,27)( 4,28)( 5,26)( 6,25)( 7,24)( 8,23)( 9,21)(10,22)(11,19)(12,20)(13,18) (14,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3, 8,19,28,24,12)( 4, 7,20,27,23,11)( 5,14, 9,25,18,22) ( 6,13,10,26,17,21)(15,16)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3,12,24,28,19, 8)( 4,11,23,27,20, 7)( 5,22,18,25, 9,14) ( 6,21,17,26,10,13)(15,16)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3,20,24, 4,19,23)( 5,10,18, 6, 9,17)( 7,28,11, 8,27,12) (13,25,21,14,26,22)(15,16)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3,23,19, 4,24,20)( 5,17, 9, 6,18,10)( 7,12,27, 8,11,28) (13,22,26,14,21,25)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1, 2)( 3,28)( 4,27)( 5,25)( 6,26)( 7,23)( 8,24)( 9,22)(10,21)(11,20)(12,19) (13,17)(14,18)(15,16)$
$ 4, 4, 4, 4, 4, 4, 4 $ $7$ $4$ $( 1, 3, 2, 4)( 5,27, 6,28)( 7,25, 8,26)( 9,24,10,23)(11,22,12,21)(13,19,14,20) (15,18,16,17)$
$ 28 $ $6$ $28$ $( 1, 3, 6, 8, 9,11,14,15,18,19,22,23,26,27, 2, 4, 5, 7,10,12,13,16,17,20,21, 24,25,28)$
$ 12, 12, 4 $ $7$ $12$ $( 1, 3,10,28,26,19, 2, 4, 9,27,25,20)( 5,16,17,23,13,11, 6,15,18,24,14,12) ( 7,22, 8,21)$
$ 12, 12, 4 $ $7$ $12$ $( 1, 3,14, 8, 5,24, 2, 4,13, 7, 6,23)( 9,16,17,28,21,19,10,15,18,27,22,20) (11,25,12,26)$
$ 12, 12, 4 $ $7$ $12$ $( 1, 3,22,15,18, 7, 2, 4,21,16,17, 8)( 5,11,10,20,26,24, 6,12, 9,19,25,23) (13,27,14,28)$
$ 12, 12, 4 $ $7$ $12$ $( 1, 3,25,15,18,11, 2, 4,26,16,17,12)( 5,19, 6,20)( 7,14,23,21,27,10, 8,13,24, 22,28, 9)$
$ 4, 4, 4, 4, 4, 4, 4 $ $7$ $4$ $( 1, 4, 2, 3)( 5,28, 6,27)( 7,26, 8,25)( 9,23,10,24)(11,21,12,22)(13,20,14,19) (15,17,16,18)$
$ 28 $ $6$ $28$ $( 1, 4, 6, 7, 9,12,14,16,18,20,22,24,26,28, 2, 3, 5, 8,10,11,13,15,17,19,21, 23,25,27)$
$ 12, 12, 4 $ $7$ $12$ $( 1, 4,10,27,26,20, 2, 3, 9,28,25,19)( 5,15,17,24,13,12, 6,16,18,23,14,11) ( 7,21, 8,22)$
$ 12, 12, 4 $ $7$ $12$ $( 1, 4,14, 7, 5,23, 2, 3,13, 8, 6,24)( 9,15,17,27,21,20,10,16,18,28,22,19) (11,26,12,25)$
$ 12, 12, 4 $ $7$ $12$ $( 1, 4,22,16,18, 8, 2, 3,21,15,17, 7)( 5,12,10,19,26,23, 6,11, 9,20,25,24) (13,28,14,27)$
$ 12, 12, 4 $ $7$ $12$ $( 1, 4,25,16,18,12, 2, 3,26,15,17,11)( 5,20, 6,19)( 7,13,23,22,27, 9, 8,14,24, 21,28,10)$
$ 7, 7, 7, 7 $ $6$ $7$ $( 1, 5, 9,13,18,21,26)( 2, 6,10,14,17,22,25)( 3, 7,11,16,19,24,27) ( 4, 8,12,15,20,23,28)$
$ 14, 14 $ $6$ $14$ $( 1, 6, 9,14,18,22,26, 2, 5,10,13,17,21,25)( 3, 8,11,15,19,23,27, 4, 7,12,16, 20,24,28)$
$ 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,15, 2,16)( 3,18, 4,17)( 5,20, 6,19)( 7,21, 8,22)( 9,23,10,24)(11,26,12,25) (13,28,14,27)$
$ 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,16, 2,15)( 3,17, 4,18)( 5,19, 6,20)( 7,22, 8,21)( 9,24,10,23)(11,25,12,26) (13,27,14,28)$

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 8]
Character table: Data not available.