Properties

Label 21T29
Order \(1764\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $29$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,3)(2,6,7)(8,15,11,16,9,20)(10,18,12,21,13,19)(14,17), (1,15,5,17,6,21)(2,19,7,18,3,16)(4,20)(8,12,14)(10,13,11), (1,4,6,5,2,7)(8,19,10,20,14,15)(9,16,12,21,11,17)(13,18)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
12:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
36:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

14T37, 21T29, 28T170, 42T223 x 2, 42T224 x 2, 42T225 x 2, 42T252, 42T253, 42T254, 42T255

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, 7 $ $18$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $18$ $7$ $( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $12$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $49$ $3$ $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $49$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$
$ 3, 3, 3, 3, 3, 3, 3 $ $98$ $3$ $( 1,14,21)( 2,13,15)( 3,12,16)( 4,11,17)( 5,10,18)( 6, 9,19)( 7, 8,20)$
$ 21 $ $84$ $21$ $( 1,11,16, 2,14,18, 3,10,20, 4,13,15, 5, 9,17, 6,12,19, 7, 8,21)$
$ 3, 3, 3, 3, 3, 3, 3 $ $14$ $3$ $( 1,13,20)( 2, 9,15)( 3,12,17)( 4, 8,19)( 5,11,21)( 6,14,16)( 7,10,18)$
$ 21 $ $84$ $21$ $( 1,13,15, 3, 9,16, 5,12,17, 7, 8,18, 2,11,19, 4,14,20, 6,10,21)$
$ 3, 3, 3, 3, 3, 3, 3 $ $14$ $3$ $( 1,10,17)( 2, 8,21)( 3,13,18)( 4,11,15)( 5, 9,19)( 6,14,16)( 7,12,20)$
$ 6, 6, 3, 3, 2, 1 $ $147$ $6$ $( 2, 3, 5)( 4, 7, 6)( 8,20,10,16,11,21)( 9,18,14,15,13,17)(12,19)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,16)(14,15)$
$ 14, 7 $ $126$ $14$ $( 1, 7, 6, 5, 4, 3, 2)( 8,18,14,19,13,20,12,21,11,15,10,16, 9,17)$
$ 6, 6, 3, 3, 2, 1 $ $147$ $6$ $( 2, 5, 3)( 4, 6, 7)( 8,18,13,19, 9,21)(10,17)(11,20,12,16,14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $49$ $2$ $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$
$ 6, 6, 6, 1, 1, 1 $ $49$ $6$ $( 2, 4, 3, 7, 5, 6)( 9,11,10,14,12,13)(16,18,17,21,19,20)$
$ 6, 6, 6, 1, 1, 1 $ $49$ $6$ $( 2, 6, 5, 7, 3, 4)( 9,13,12,14,10,11)(16,20,19,21,17,18)$
$ 6, 6, 6, 3 $ $98$ $6$ $( 1,14,19, 5,11,16)( 2, 8,20, 4,10,15)( 3, 9,21)( 6,12,17, 7,13,18)$
$ 6, 6, 6, 3 $ $98$ $6$ $( 1,11,19, 3,12,16)( 2, 8,21)( 4, 9,18, 7,14,17)( 5,13,20, 6,10,15)$
$ 6, 6, 6, 3 $ $98$ $6$ $( 1,13,21, 5,14,16)( 2, 8,18, 4,12,19)( 3,10,15)( 6, 9,20, 7,11,17)$
$ 6, 6, 6, 2, 1 $ $147$ $6$ $( 2, 6, 5, 7, 3, 4)( 8,20, 9,15,13,16)(10,17)(11,19,14,18,12,21)$
$ 14, 2, 2, 2, 1 $ $126$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,21,13,19,11,17, 9,15,14,20,12,18,10,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $21$ $2$ $( 2, 7)( 3, 6)( 4, 5)( 8,15)( 9,16)(10,17)(11,18)(12,19)(13,20)(14,21)$
$ 6, 6, 6, 2, 1 $ $147$ $6$ $( 2, 4, 3, 7, 5, 6)( 8,18, 9,15,11,16)(10,19,13,17,12,20)(14,21)$

Group invariants

Order:  $1764=2^{2} \cdot 3^{2} \cdot 7^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1764, 134]
Character table: Data not available.