Properties

Label 21T9
Order \(126\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times F_7$

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $9$
Group :  $C_3\times F_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,2,14,3,15)(4,10,5,11,6,12)(7,9,8)(16,19,17,20,18,21), (1,12,8)(2,10,9)(3,11,7)(4,16,20)(5,17,21)(6,18,19)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$ x 4
6:  $C_6$ x 4
9:  $C_3^2$
18:  $C_6 \times C_3$
42:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $F_7$

Low degree siblings

21T9 x 2, 42T17 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $7$ $3$ $( 4, 7,13)( 5, 8,14)( 6, 9,15)(10,19,18)(11,20,16)(12,21,17)$
$ 6, 6, 6, 1, 1, 1 $ $7$ $6$ $( 4,11, 7,20,13,16)( 5,12, 8,21,14,17)( 6,10, 9,19,15,18)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $7$ $3$ $( 4,13, 7)( 5,14, 8)( 6,15, 9)(10,18,19)(11,16,20)(12,17,21)$
$ 6, 6, 6, 1, 1, 1 $ $7$ $6$ $( 4,16,13,20, 7,11)( 5,17,14,21, 8,12)( 6,18,15,19, 9,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $7$ $2$ $( 4,20)( 5,21)( 6,19)( 7,16)( 8,17)( 9,18)(10,15)(11,13)(12,14)$
$ 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1, 2, 3)( 4, 8,15)( 5, 9,13)( 6, 7,14)(10,20,17)(11,21,18)(12,19,16)$
$ 6, 6, 6, 3 $ $7$ $6$ $( 1, 2, 3)( 4,12, 9,20,14,18)( 5,10, 7,21,15,16)( 6,11, 8,19,13,17)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1, 2, 3)( 4,14, 9)( 5,15, 7)( 6,13, 8)(10,16,21)(11,17,19)(12,18,20)$
$ 6, 6, 6, 3 $ $7$ $6$ $( 1, 2, 3)( 4,17,15,20, 8,10)( 5,18,13,21, 9,11)( 6,16,14,19, 7,12)$
$ 6, 6, 6, 3 $ $7$ $6$ $( 1, 2, 3)( 4,21, 6,20, 5,19)( 7,17, 9,16, 8,18)(10,13,12,15,11,14)$
$ 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1, 3, 2)( 4, 9,14)( 5, 7,15)( 6, 8,13)(10,21,16)(11,19,17)(12,20,18)$
$ 6, 6, 6, 3 $ $7$ $6$ $( 1, 3, 2)( 4,10, 8,20,15,17)( 5,11, 9,21,13,18)( 6,12, 7,19,14,16)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1, 3, 2)( 4,15, 8)( 5,13, 9)( 6,14, 7)(10,17,20)(11,18,21)(12,16,19)$
$ 6, 6, 6, 3 $ $7$ $6$ $( 1, 3, 2)( 4,18,14,20, 9,12)( 5,16,15,21, 7,10)( 6,17,13,19, 8,11)$
$ 6, 6, 6, 3 $ $7$ $6$ $( 1, 3, 2)( 4,19, 5,20, 6,21)( 7,18, 8,16, 9,17)(10,14,11,15,12,13)$
$ 21 $ $6$ $21$ $( 1, 4, 9,12,13,18,21, 3, 6, 8,11,15,17,20, 2, 5, 7,10,14,16,19)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 5, 8,12,14,17,21)( 2, 6, 9,10,15,18,19)( 3, 4, 7,11,13,16,20)$
$ 21 $ $6$ $21$ $( 1, 6, 7,12,15,16,21, 2, 4, 8,10,13,17,19, 3, 5, 9,11,14,18,20)$

Group invariants

Order:  $126=2 \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [126, 7]
Character table: Data not available.