Properties

Label 42T27
Order \(168\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_7:A_4$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $27$
Group :  $C_2\times C_7:A_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,35,21,7,38,28,13)(2,36,22,8,37,27,14)(3,31,24,9,40,30,16,4,32,23,10,39,29,15)(5,33,20,12,42,25,17,6,34,19,11,41,26,18), (1,23,17,2,24,18)(3,19,13,4,20,14)(5,22,16,6,21,15)(7,31,42,8,32,41)(9,34,37,10,33,38)(11,36,40,12,35,39)(25,28,30,26,27,29)
$|\Aut(F/K)|$:  $14$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
21:  $C_7:C_3$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4\times C_2$

Degree 7: $C_7:C_3$

Degree 14: None

Degree 21: 21T2

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 53]
Character table: Data not available.