# Properties

 Label 42T13 Order $$84$$ n $$42$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $S_3\times D_7$

## Group action invariants

 Degree $n$ : $42$ Transitive number $t$ : $13$ Group : $S_3\times D_7$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,27,7,34,13,39,19,4,25,10,31,16,37,21)(2,28,8,33,14,40,20,3,26,9,32,15,38,22)(5,30,11,36,17,41,23,6,29,12,35,18,42,24), (3,5)(4,6)(7,37)(8,38)(9,42)(10,41)(11,40)(12,39)(13,31)(14,32)(15,35)(16,36)(17,33)(18,34)(19,25)(20,26)(21,30)(22,29)(23,28)(24,27) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
14:  $D_{7}$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 7: $D_{7}$

Degree 14: $D_{14}$

Degree 21: 21T8

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

## Group invariants

 Order: $84=2^{2} \cdot 3 \cdot 7$ Cyclic: No Abelian: No Solvable: Yes GAP id: [84, 8]
 Character table:  2 2 2 2 2 1 1 1 1 . 1 1 . 1 1 . 3 1 . 1 . 1 1 1 . 1 1 . 1 1 . 1 7 1 . . 1 1 . 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 3a 6a 7a 14a 21a 7b 14b 21b 7c 14c 21c 2P 1a 1a 1a 1a 3a 3a 7b 7b 21b 7c 7c 21c 7a 7a 21a 3P 1a 2a 2b 2c 1a 2b 7c 14c 7c 7a 14a 7a 7b 14b 7b 5P 1a 2a 2b 2c 3a 6a 7b 14b 21b 7c 14c 21c 7a 14a 21a 7P 1a 2a 2b 2c 3a 6a 1a 2c 3a 1a 2c 3a 1a 2c 3a 11P 1a 2a 2b 2c 3a 6a 7c 14c 21c 7a 14a 21a 7b 14b 21b 13P 1a 2a 2b 2c 3a 6a 7a 14a 21a 7b 14b 21b 7c 14c 21c 17P 1a 2a 2b 2c 3a 6a 7c 14c 21c 7a 14a 21a 7b 14b 21b 19P 1a 2a 2b 2c 3a 6a 7b 14b 21b 7c 14c 21c 7a 14a 21a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 1 1 1 1 1 1 1 1 1 X.3 1 -1 1 -1 1 1 1 -1 1 1 -1 1 1 -1 1 X.4 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 X.5 2 . -2 . -1 1 2 . -1 2 . -1 2 . -1 X.6 2 . 2 . -1 -1 2 . -1 2 . -1 2 . -1 X.7 2 . . -2 2 . A -A A C -C C B -B B X.8 2 . . -2 2 . B -B B A -A A C -C C X.9 2 . . -2 2 . C -C C B -B B A -A A X.10 2 . . 2 2 . A A A C C C B B B X.11 2 . . 2 2 . B B B A A A C C C X.12 2 . . 2 2 . C C C B B B A A A X.13 4 . . . -2 . D . -B F . -A E . -C X.14 4 . . . -2 . E . -C D . -B F . -A X.15 4 . . . -2 . F . -A E . -C D . -B A = E(7)^3+E(7)^4 B = E(7)^2+E(7)^5 C = E(7)+E(7)^6 D = 2*E(7)^2+2*E(7)^5 E = 2*E(7)+2*E(7)^6 F = 2*E(7)^3+2*E(7)^4