Properties

Label 28T31
Order \(168\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7\times S_4$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $31$
Group :  $C_7\times S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,24,15,6,25,20,11,2,21,16,7,26,17,12,3,22,13,8,27,18,9,4,23,14,5,28,19,10), (1,27,24,17,15,12,5,3,28,21,19,16,9,7,4,25,23,20,13,11,8)(2,26,22,18,14,10,6)
$|\Aut(F/K)|$:  $7$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
7:  $C_7$
14:  $C_{14}$
24:  $S_4$
42:  21T6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Degree 7: $C_7$

Degree 14: None

Low degree siblings

42T34, 42T35

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 45]
Character table: Data not available.