Properties

Label 28T24
Order \(168\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times F_7$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $24$
Group :  $C_2^2\times F_7$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,18,9,26,21)(2,14,17,10,25,22)(3,24,11,7,16,27)(4,23,12,8,15,28), (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), (1,16)(2,15)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
3:  $C_3$
4:  $C_2^2$ x 7
6:  $C_6$ x 7
8:  $C_2^3$
12:  $C_6\times C_2$ x 7
24:  24T3
42:  $F_7$
84:  $F_7 \times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $F_7$

Degree 14: $F_7 \times C_2$ x 3

Low degree siblings

28T24 x 3

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 47]
Character table: Data not available.