Properties

Label 42T21
Order \(126\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times D_{21}$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $21$
Group :  $C_3\times D_{21}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,36,2,34,3,35)(4,33,5,31,6,32)(7,29,8,30,9,28)(10,26,12,25,11,27)(13,24,15,23,14,22)(16,21,18,20,17,19)(37,41,38,42,39,40), (1,16,2,18,3,17)(4,13,5,15,6,14)(7,12,8,11,9,10)(19,41,21,42,20,40)(22,38,24,39,23,37)(25,35,27,36,26,34)(28,32,29,33,30,31)
$|\Aut(F/K)|$:  $21$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
14:  $D_{7}$
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $S_3\times C_3$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

Degree 21: None

Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $3$ $( 4, 5, 6)(10,12,11)(16,18,17)(22,24,23)(28,29,30)(34,35,36)(40,41,42)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $3$ $( 4, 6, 5)(10,11,12)(16,17,18)(22,23,24)(28,30,29)(34,36,35)(40,42,41)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,17,18)(19,21,20) (22,23,24)(25,27,26)(28,30,29)(31,32,33)(34,36,35)(37,38,39)(40,42,41)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 4)( 2, 5)( 3, 6)( 7,42)( 8,40)( 9,41)(10,39)(11,38)(12,37)(13,35)(14,34) (15,36)(16,31)(17,33)(18,32)(19,29)(20,28)(21,30)(22,27)(23,25)(24,26)$
$ 6, 6, 6, 6, 6, 6, 6 $ $21$ $6$ $( 1, 4, 2, 5, 3, 6)( 7,42, 8,40, 9,41)(10,37,12,38,11,39)(13,35,15,36,14,34) (16,32,18,33,17,31)(19,29,21,30,20,28)(22,26,24,25,23,27)$
$ 6, 6, 6, 6, 6, 6, 6 $ $21$ $6$ $( 1, 4, 3, 6, 2, 5)( 7,42, 9,41, 8,40)(10,38,11,37,12,39)(13,35,14,34,15,36) (16,33,17,32,18,31)(19,29,20,28,21,30)(22,25,23,26,24,27)$
$ 21, 21 $ $2$ $21$ $( 1, 7,13,20,27,32,38, 3, 9,14,21,25,31,37, 2, 8,15,19,26,33,39) ( 4,10,17,24,29,36,40, 5,12,16,23,30,34,41, 6,11,18,22,28,35,42)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1, 7,13,20,27,32,38, 3, 9,14,21,25,31,37, 2, 8,15,19,26,33,39) ( 4,11,16,24,28,34,40)( 5,10,18,23,29,35,41)( 6,12,17,22,30,36,42)$
$ 21, 21 $ $2$ $21$ $( 1, 7,13,20,27,32,38, 3, 9,14,21,25,31,37, 2, 8,15,19,26,33,39) ( 4,12,18,24,30,35,40, 6,10,16,22,29,34,42, 5,11,17,23,28,36,41)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1, 8,14,20,26,31,38)( 2, 9,13,19,25,32,39)( 3, 7,15,21,27,33,37) ( 4,10,17,24,29,36,40, 5,12,16,23,30,34,41, 6,11,18,22,28,35,42)$
$ 7, 7, 7, 7, 7, 7 $ $2$ $7$ $( 1, 8,14,20,26,31,38)( 2, 9,13,19,25,32,39)( 3, 7,15,21,27,33,37) ( 4,11,16,24,28,34,40)( 5,10,18,23,29,35,41)( 6,12,17,22,30,36,42)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1, 8,14,20,26,31,38)( 2, 9,13,19,25,32,39)( 3, 7,15,21,27,33,37) ( 4,12,18,24,30,35,40, 6,10,16,22,29,34,42, 5,11,17,23,28,36,41)$
$ 21, 21 $ $2$ $21$ $( 1, 9,15,20,25,33,38, 2, 7,14,19,27,31,39, 3, 8,13,21,26,32,37) ( 4,10,17,24,29,36,40, 5,12,16,23,30,34,41, 6,11,18,22,28,35,42)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1, 9,15,20,25,33,38, 2, 7,14,19,27,31,39, 3, 8,13,21,26,32,37) ( 4,11,16,24,28,34,40)( 5,10,18,23,29,35,41)( 6,12,17,22,30,36,42)$
$ 21, 21 $ $2$ $21$ $( 1, 9,15,20,25,33,38, 2, 7,14,19,27,31,39, 3, 8,13,21,26,32,37) ( 4,12,18,24,30,35,40, 6,10,16,22,29,34,42, 5,11,17,23,28,36,41)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1,13,27,38, 9,21,31, 2,15,26,39, 7,20,32, 3,14,25,37, 8,19,33) ( 4,16,28,40,11,24,34)( 5,18,29,41,10,23,35)( 6,17,30,42,12,22,36)$
$ 21, 21 $ $2$ $21$ $( 1,13,27,38, 9,21,31, 2,15,26,39, 7,20,32, 3,14,25,37, 8,19,33) ( 4,17,29,40,12,23,34, 6,18,28,42,10,24,36, 5,16,30,41,11,22,35)$
$ 21, 21 $ $2$ $21$ $( 1,13,27,38, 9,21,31, 2,15,26,39, 7,20,32, 3,14,25,37, 8,19,33) ( 4,18,30,40,10,22,34, 5,17,28,41,12,24,35, 6,16,29,42,11,23,36)$
$ 7, 7, 7, 7, 7, 7 $ $2$ $7$ $( 1,14,26,38, 8,20,31)( 2,13,25,39, 9,19,32)( 3,15,27,37, 7,21,33) ( 4,16,28,40,11,24,34)( 5,18,29,41,10,23,35)( 6,17,30,42,12,22,36)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1,14,26,38, 8,20,31)( 2,13,25,39, 9,19,32)( 3,15,27,37, 7,21,33) ( 4,17,29,40,12,23,34, 6,18,28,42,10,24,36, 5,16,30,41,11,22,35)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1,14,26,38, 8,20,31)( 2,13,25,39, 9,19,32)( 3,15,27,37, 7,21,33) ( 4,18,30,40,10,22,34, 5,17,28,41,12,24,35, 6,16,29,42,11,23,36)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1,15,25,38, 7,19,31, 3,13,26,37, 9,20,33, 2,14,27,39, 8,21,32) ( 4,16,28,40,11,24,34)( 5,18,29,41,10,23,35)( 6,17,30,42,12,22,36)$
$ 21, 21 $ $2$ $21$ $( 1,15,25,38, 7,19,31, 3,13,26,37, 9,20,33, 2,14,27,39, 8,21,32) ( 4,17,29,40,12,23,34, 6,18,28,42,10,24,36, 5,16,30,41,11,22,35)$
$ 21, 21 $ $2$ $21$ $( 1,15,25,38, 7,19,31, 3,13,26,37, 9,20,33, 2,14,27,39, 8,21,32) ( 4,18,30,40,10,22,34, 5,17,28,41,12,24,35, 6,16,29,42,11,23,36)$
$ 21, 21 $ $2$ $21$ $( 1,19,37,14,32, 7,26, 2,21,38,13,33, 8,25, 3,20,39,15,31, 9,27) ( 4,22,41,16,36,10,28, 6,23,40,17,35,11,30, 5,24,42,18,34,12,29)$
$ 21, 21 $ $2$ $21$ $( 1,19,37,14,32, 7,26, 2,21,38,13,33, 8,25, 3,20,39,15,31, 9,27) ( 4,23,42,16,35,12,28, 5,22,40,18,36,11,29, 6,24,41,17,34,10,30)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1,19,37,14,32, 7,26, 2,21,38,13,33, 8,25, 3,20,39,15,31, 9,27) ( 4,24,40,16,34,11,28)( 5,23,41,18,35,10,29)( 6,22,42,17,36,12,30)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1,20,38,14,31, 8,26)( 2,19,39,13,32, 9,25)( 3,21,37,15,33, 7,27) ( 4,22,41,16,36,10,28, 6,23,40,17,35,11,30, 5,24,42,18,34,12,29)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1,20,38,14,31, 8,26)( 2,19,39,13,32, 9,25)( 3,21,37,15,33, 7,27) ( 4,23,42,16,35,12,28, 5,22,40,18,36,11,29, 6,24,41,17,34,10,30)$
$ 7, 7, 7, 7, 7, 7 $ $2$ $7$ $( 1,20,38,14,31, 8,26)( 2,19,39,13,32, 9,25)( 3,21,37,15,33, 7,27) ( 4,24,40,16,34,11,28)( 5,23,41,18,35,10,29)( 6,22,42,17,36,12,30)$
$ 21, 21 $ $2$ $21$ $( 1,21,39,14,33, 9,26, 3,19,38,15,32, 8,27, 2,20,37,13,31, 7,25) ( 4,22,41,16,36,10,28, 6,23,40,17,35,11,30, 5,24,42,18,34,12,29)$
$ 21, 21 $ $2$ $21$ $( 1,21,39,14,33, 9,26, 3,19,38,15,32, 8,27, 2,20,37,13,31, 7,25) ( 4,23,42,16,35,12,28, 5,22,40,18,36,11,29, 6,24,41,17,34,10,30)$
$ 21, 7, 7, 7 $ $2$ $21$ $( 1,21,39,14,33, 9,26, 3,19,38,15,32, 8,27, 2,20,37,13,31, 7,25) ( 4,24,40,16,34,11,28)( 5,23,41,18,35,10,29)( 6,22,42,17,36,12,30)$

Group invariants

Order:  $126=2 \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [126, 13]
Character table: Data not available.