Properties

Label 42T36
Order \(168\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_3\times D_7$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $36$
Group :  $C_2\times S_3\times D_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,37,13,31,8,25)(2,20,38,14,32,7,26)(3,23,40,18,33,11,27,6,21,41,16,35,10,30)(4,24,39,17,34,12,28,5,22,42,15,36,9,29), (1,23)(2,24)(3,22)(4,21)(5,20)(6,19)(7,17)(8,18)(9,16)(10,15)(11,13)(12,14)(25,41)(26,42)(27,39)(28,40)(29,38)(30,37)(31,35)(32,36)(33,34), (1,11,15,20,29,33,37,6,9,14,24,27,31,41,4,7,17,21,25,35,39,2,12,16,19,30,34,38,5,10,13,23,28,32,42,3,8,18,22,26,36,40)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $C_2^3$
12:  $D_{6}$ x 3
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: 21T8

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 50]
Character table: Data not available.