Properties

Label 42T95
Degree $42$
Order $504$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times S_3\times F_7$

Related objects

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Group action invariants

Degree $n$:  $42$
Transitive number $t$:  $95$
Group:  $C_2\times S_3\times F_7$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $2$
Generators:  (1,17,20,5,13,23)(2,18,19,6,14,24)(3,15,21,4,16,22)(7,42,32,11,38,36)(8,41,31,12,37,35)(9,39,33,10,40,34)(25,29)(26,30)(27,28), (1,39)(2,40)(3,38)(4,37)(5,41)(6,42)(7,33)(8,34)(9,32)(10,31)(11,36)(12,35)(13,28)(14,27)(15,26)(16,25)(17,30)(18,29)(19,21)(20,22), (1,16,42,2,15,41)(3,18,38,4,17,37)(5,13,40,6,14,39)(7,28,23,8,27,24)(9,29,19,10,30,20)(11,25,22,12,26,21)(31,33,36,32,34,35)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$3$:  $C_3$
$4$:  $C_2^2$ x 7
$6$:  $S_3$, $C_6$ x 7
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$18$:  $S_3\times C_3$
$42$:  $F_7$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 7: $F_7$

Degree 14: $F_7 \times C_2$

Degree 21: 21T15

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

Group invariants

Order:  $504=2^{3} \cdot 3^{2} \cdot 7$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [504, 162]
Character table: not available.