Properties

Label 42T44
Order \(252\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times F_7$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $44$
Group :  $S_3\times F_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,26,41,32,18)(2,12,25,42,31,17)(3,9,28,39,34,15)(4,10,27,40,33,16)(5,7,29,38,35,14)(6,8,30,37,36,13)(19,23)(20,24)(21,22), (1,21,13,3,20,16)(2,22,14,4,19,15)(5,23,17)(6,24,18)(7,33,38,9,31,39)(8,34,37,10,32,40)(11,36,41)(12,35,42)(25,27)(26,28)
$|\Aut(F/K)|$:  $2$

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:  $S_3\times C_3$
42:  $F_7$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 7: $F_7$

Degree 14: $F_7$

Degree 21: 21T15

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

Group invariants

Order:  $252=2^{2} \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [252, 26]
Character table: Data not available.