Properties

Label 42T47
Order \(252\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7\times C_3:S_3.C_2$

Learn more about

Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $47$
Group :  $C_7\times C_3:S_3.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3)(4,6,5)(7,9,8)(10,11,12)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)(40,42,41), (1,17,31,6,19,35,7,24,38,10,27,40,14,28,2,16,32,5,21,36,8,22,37,12,25,42,13,30)(3,18,33,4,20,34,9,23,39,11,26,41,15,29)
$|\Aut(F/K)|$:  $7$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
7:  $C_7$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$

Degree 7: $C_7$

Degree 14: $C_{14}$

Degree 21: None

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

Group invariants

Order:  $252=2^{2} \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [252, 31]
Character table: Data not available.