Properties

Label 42T42
Order \(252\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_3\times C_7:C_3$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $42$
Group :  $C_2\times S_3\times C_7:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,14,21,25,34,37,3,7,16,19,27,32,40)(2,10,13,22,26,33,38,4,8,15,20,28,31,39)(5,12,18,23,29,36,42,6,11,17,24,30,35,41), (1,12,22,2,11,21)(3,7,23,4,8,24)(5,9,19,6,10,20)(13,35,27,14,36,28)(15,31,29,16,32,30)(17,33,26,18,34,25)(37,41,39,38,42,40)
$|\Aut(F/K)|$:  $2$

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:  $D_{6}$
18:  $S_3\times C_3$
21:  $C_7:C_3$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 7: $C_7:C_3$

Degree 14: $(C_7:C_3) \times C_2$

Degree 21: 21T11

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

Group invariants

Order:  $252=2^{2} \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [252, 29]
Character table: Data not available.