# Properties

 Label 42T30 Order $$168$$ n $$42$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_7:A_4$

## Group action invariants

 Degree $n$ : $42$ Transitive number $t$ : $30$ Group : $D_7:A_4$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,23,37,15,36,8,29,2,24,38,16,35,7,30)(3,22,39,13,31,10,27)(4,21,40,14,32,9,28)(5,20,41,17,33,11,25,6,19,42,18,34,12,26), (1,22,26,15,40,33)(2,21,25,16,39,34)(3,20,30,14,41,36)(4,19,29,13,42,35)(5,24,27,17,38,32)(6,23,28,18,37,31)(7,10,11,8,9,12) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$
42:  $F_7$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4\times C_2$

Degree 7: $F_7$

Degree 14: None

Degree 21: 21T4

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

## Group invariants

 Order: $168=2^{3} \cdot 3 \cdot 7$ Cyclic: No Abelian: No Solvable: Yes GAP id: [168, 49]
 Character table:  2 3 3 3 3 1 1 1 1 2 2 2 2 3 1 . . 1 1 1 1 1 . . . . 7 1 . 1 . . . . . 1 1 1 1 1a 2a 2b 2c 3a 6a 3b 6b 14a 7a 14b 14c 2P 1a 1a 1a 1a 3b 3b 3a 3a 7a 7a 7a 7a 3P 1a 2a 2b 2c 1a 2c 1a 2c 14c 7a 14a 14b 5P 1a 2a 2b 2c 3b 6b 3a 6a 14b 7a 14c 14a 7P 1a 2a 2b 2c 3a 6a 3b 6b 2b 1a 2b 2b 11P 1a 2a 2b 2c 3b 6b 3a 6a 14c 7a 14a 14b 13P 1a 2a 2b 2c 3a 6a 3b 6b 14a 7a 14b 14c X.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 X.3 1 -1 1 -1 A -A /A -/A 1 1 1 1 X.4 1 -1 1 -1 /A -/A A -A 1 1 1 1 X.5 1 1 1 1 A A /A /A 1 1 1 1 X.6 1 1 1 1 /A /A A A 1 1 1 1 X.7 3 -1 -1 3 . . . . -1 3 -1 -1 X.8 3 1 -1 -3 . . . . -1 3 -1 -1 X.9 6 . 6 . . . . . -1 -1 -1 -1 X.10 6 . -2 . . . . . B -1 D C X.11 6 . -2 . . . . . C -1 B D X.12 6 . -2 . . . . . D -1 C B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(7)-E(7)^2-E(7)^3-E(7)^4-E(7)^5+E(7)^6 C = -E(7)-E(7)^2+E(7)^3+E(7)^4-E(7)^5-E(7)^6 D = -E(7)+E(7)^2-E(7)^3-E(7)^4+E(7)^5-E(7)^6