Properties

Label 42T16
Order \(126\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_6\times C_7:C_3$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $16$
Group :  $C_6\times C_7:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,39,2,18,40)(3,13,41,4,14,42)(5,16,38,6,15,37)(7,28,20,8,27,19)(9,30,22,10,29,21)(11,25,24,12,26,23)(31,32)(33,34)(35,36), (1,22,11,2,21,12)(3,24,7,4,23,8)(5,19,9,6,20,10)(13,25,36,14,26,35)(15,28,31,16,27,32)(17,30,33,18,29,34)(37,40,42,38,39,41)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$ x 4
6:  $C_6$ x 4
9:  $C_3^2$
21:  $C_7:C_3$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 7: $C_7:C_3$

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

Degree 21: 21T7

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

Group invariants

Order:  $126=2 \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [126, 10]
Character table: Data not available.