# Properties

 Label 42T8 Degree $42$ Order $84$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_7:A_4$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$12$:  $A_4$
$21$:  $C_7:C_3$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4$

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

## Group invariants

 Order: $84=2^{2} \cdot 3 \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [84, 11]
 Character table:  2 2 2 . . 2 2 2 2 2 2 2 2 3 1 . 1 1 . . . . . . . . 7 1 1 . . 1 1 1 1 1 1 1 1 1a 2a 3a 3b 7a 14a 14b 14c 14d 14e 7b 14f 2P 1a 1a 3b 3a 7a 7a 7a 7a 7b 7b 7b 7b 3P 1a 2a 1a 1a 7b 14f 14d 14e 14a 14b 7a 14c 5P 1a 2a 3b 3a 7b 14d 14e 14f 14b 14c 7a 14a 7P 1a 2a 3a 3b 1a 2a 2a 2a 2a 2a 1a 2a 11P 1a 2a 3b 3a 7a 14b 14c 14a 14e 14f 7b 14d 13P 1a 2a 3a 3b 7b 14e 14f 14d 14c 14a 7a 14b X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 A /A 1 1 1 1 1 1 1 1 X.3 1 1 /A A 1 1 1 1 1 1 1 1 X.4 3 -1 . . 3 -1 -1 -1 -1 -1 3 -1 X.5 3 3 . . B B B B /B /B /B /B X.6 3 3 . . /B /B /B /B B B B B X.7 3 -1 . . B C D E /E /C /B /D X.8 3 -1 . . B D E C /C /D /B /E X.9 3 -1 . . B E C D /D /E /B /C X.10 3 -1 . . /B /E /C /D D E B C X.11 3 -1 . . /B /C /D /E E C B D X.12 3 -1 . . /B /D /E /C C D B E A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 C = -E(7)^3-E(7)^5+E(7)^6 D = E(7)^3-E(7)^5-E(7)^6 E = -E(7)^3+E(7)^5-E(7)^6