Properties

Label 42T1
Order \(42\)
n \(42\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive No
$p$-group No
Group: $C_{42}$

Related objects

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $1$
Group :  $C_{42}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,21,38,17,35,11,27,3,23,40,13,31,8,29,6,20,42,15,34,10,25,2,22,37,18,36,12,28,4,24,39,14,32,7,30,5,19,41,16,33,9,26)
$|\Aut(F/K)|$:  $42$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
7:  $C_7$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 7: $C_7$

Degree 14: $C_{14}$

Degree 21: $C_{21}$

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

Group invariants

Order:  $42=2 \cdot 3 \cdot 7$
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [42, 6]
Character table: Data not available.