Properties

Label 42T50
Order \(252\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times S_3\times D_7$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $50$
Group :  $C_3\times S_3\times D_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,40,39,34,31,30,27,22,19,17,13,11,9,4,3,41,38,36,32,29,25,24,20,16,15,12,8,5,2,42,37,35,33,28,26,23,21,18,14,10,7,6), (1,5,3,6,2,4)(7,40,9,41,8,42)(10,39,11,38,12,37)(13,35,15,34,14,36)(16,31,18,32,17,33)(19,30,20,29,21,28)(22,27,24,25,23,26)
$|\Aut(F/K)|$:  $3$

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:  $D_{6}$
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: $D_{14}$

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

Group invariants

Order:  $252=2^{2} \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [252, 33]
Character table: Data not available.