Properties

Label 21T16
Order \(294\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7:(C_3\times D_7)$

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $16$
Group :  $C_7:(C_3\times D_7)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,20,2,12,17)(3,10,21,7,9,16)(4,8,18,6,11,19)(5,13,15), (1,2)(3,7)(4,6)(8,12)(9,11)(13,14)(15,18)(16,17)(19,21)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
14:  $D_{7}$
42:  $F_7$, 21T3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T16, 42T55 x 2

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 $ $1$ $1$ $()$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $49$ $2$ $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $2$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $2$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $2$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,20,18,16,21,19,17)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1, 8,15)( 2,10,19)( 3,12,16)( 4,14,20)( 5, 9,17)( 6,11,21)( 7,13,18)$
$ 21 $ $14$ $21$ $( 1, 8,16, 6,11,15, 4,14,21, 2,10,20, 7,13,19, 5, 9,18, 3,12,17)$
$ 21 $ $14$ $21$ $( 1, 8,17, 4,14,15, 7,13,20, 3,12,18, 6,11,16, 2,10,21, 5, 9,19)$
$ 21 $ $14$ $21$ $( 1, 8,18, 2,10,15, 3,12,19, 4,14,16, 5, 9,20, 6,11,17, 7,13,21)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1, 8,15)( 2,13,19, 7,10,18)( 3,11,16, 6,12,21)( 4, 9,20, 5,14,17)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1,15, 8)( 2,18,10, 7,19,13)( 3,21,12, 6,16,11)( 4,17,14, 5,20, 9)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1,15, 8)( 2,19,10)( 3,16,12)( 4,20,14)( 5,17, 9)( 6,21,11)( 7,18,13)$
$ 21 $ $14$ $21$ $( 1,15,11, 7,18, 9, 6,21,14, 5,17,12, 4,20,10, 3,16, 8, 2,19,13)$
$ 21 $ $14$ $21$ $( 1,15,14, 6,21,10, 4,20,13, 2,19, 9, 7,18,12, 5,17, 8, 3,16,11)$
$ 21 $ $14$ $21$ $( 1,15,10, 5,17,11, 2,19,12, 6,21,13, 3,16,14, 7,18, 8, 4,20, 9)$

Group invariants

Order:  $294=2 \cdot 3 \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [294, 10]
Character table: Data not available.