Properties

Label 42T11
Order \(84\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{42}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
14:  $D_{7}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 7: $D_{7}$

Degree 14: $D_{14}$

Degree 21: $D_{21}$

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

Group invariants

Order:  $84=2^{2} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [84, 14]
Character table: Data not available.