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Properties

Label 14T44
Order \(2688\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $44$
CHM label :  $[2^{7}]F_{21}(7)=2wrF_{21}(7)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,11)(2,4,8)(3,13,5)(6,12,10), (7,14), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
21:  $C_7:C_3$
42:  $(C_7:C_3) \times C_2$
168:  $C_2^3:(C_7: C_3)$ x 2
336:  14T18 x 2
1344:  14T35

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7:C_3$

Low degree siblings

28T226, 28T235 x 2, 28T236, 42T309, 42T310, 42T311

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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 3,10)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 2, 9)( 3,10)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 2, 9)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $7$ $2$ $( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 7, 7 $ $192$ $7$ $( 1,13,11, 9, 7, 5, 3)( 2,14,12,10, 8, 6, 4)$
$ 7, 7 $ $192$ $7$ $( 1, 9, 3,11, 5,13, 7)( 2,10, 4,12, 6,14, 8)$
$ 3, 3, 3, 3, 1, 1 $ $112$ $3$ $( 1,13, 9)( 2, 8, 6)( 4,12,14)( 5, 7,11)$
$ 6, 3, 3, 2 $ $112$ $6$ $( 1,13, 9)( 2, 8, 6)( 3,10)( 4,12, 7,11, 5,14)$
$ 6, 6, 1, 1 $ $112$ $6$ $( 1, 6, 2, 8,13, 9)( 4,12, 7,11, 5,14)$
$ 6, 3, 3, 2 $ $112$ $6$ $( 1, 6, 2, 8,13, 9)( 3,10)( 4,12,14)( 5, 7,11)$
$ 3, 3, 3, 3, 1, 1 $ $112$ $3$ $( 1, 9,13)( 2, 6, 8)( 4,14,12)( 5,11, 7)$
$ 6, 3, 3, 2 $ $112$ $6$ $( 1, 9,13)( 2, 6, 8)( 3,10)( 4, 7, 5,11,14,12)$
$ 6, 6, 1, 1 $ $112$ $6$ $( 1, 9, 6, 8, 2,13)( 4, 7, 5,11,14,12)$
$ 6, 3, 3, 2 $ $112$ $6$ $( 1, 9, 6, 8, 2,13)( 3,10)( 4,14,12)( 5,11, 7)$
$ 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 7,14)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 3,10)( 6,13)( 7,14)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 5,12)( 6,13)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 2, 9)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $21$ $2$ $( 2, 9)( 3,10)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 14 $ $192$ $14$ $( 1,13,11, 9, 7,12,10, 8, 6, 4, 2,14, 5, 3)$
$ 14 $ $192$ $14$ $( 1, 9, 3,11, 5,13, 7, 8, 2,10, 4,12, 6,14)$
$ 6, 3, 3, 1, 1 $ $112$ $6$ $( 1,13, 9)( 2, 8, 6)( 4,12,14,11, 5, 7)$
$ 3, 3, 3, 3, 2 $ $112$ $6$ $( 1,13, 9)( 2, 8, 6)( 3,10)( 4,12, 7)( 5,14,11)$
$ 6, 3, 3, 1, 1 $ $112$ $6$ $( 1, 6, 2, 8,13, 9)( 4,12, 7)( 5,14,11)$
$ 6, 6, 2 $ $112$ $6$ $( 1, 6, 2, 8,13, 9)( 3,10)( 4,12,14,11, 5, 7)$
$ 6, 3, 3, 1, 1 $ $112$ $6$ $( 1, 9,13)( 2, 6, 8)( 4,14, 5,11, 7,12)$
$ 3, 3, 3, 3, 2 $ $112$ $6$ $( 1, 9,13)( 2, 6, 8)( 3,10)( 4, 7,12)( 5,11,14)$
$ 6, 3, 3, 1, 1 $ $112$ $6$ $( 1, 9, 6, 8, 2,13)( 4, 7,12)( 5,11,14)$
$ 6, 6, 2 $ $112$ $6$ $( 1, 9, 6, 8, 2,13)( 3,10)( 4,14, 5,11, 7,12)$

Group invariants

Order:  $2688=2^{7} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.