Properties

Label 42T35
Order \(168\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7\times S_4$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $35$
Group :  $C_7\times S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,39,36,25,21,18,8,4,42,31,28,23,14,9,6,38,34,29,20,15,11)(2,40,35,26,22,17,7,3,41,32,27,24,13,10,5,37,33,30,19,16,12), (1,17,25,41,8,24,31,5,14,30,38,12,20,35)(2,18,26,42,7,23,32,6,13,29,37,11,19,36)(3,16,27,40,10,22,33)(4,15,28,39,9,21,34)
$|\Aut(F/K)|$:  $14$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
7:  $C_7$
24:  $S_4$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4$

Degree 7: $C_7$

Degree 14: None

Degree 21: 21T6

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 45]
Character table: Data not available.