Properties

 Label 42T5 Degree $42$ Order $42$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{21}$

Learn more

Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

Degree 21: $D_{21}$

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

Group invariants

 Order: $42=2 \cdot 3 \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [42, 5]
 Character table:  2 1 1 . . . . . . . . . . 3 1 . 1 1 1 1 1 1 1 1 1 1 7 1 . 1 1 1 1 1 1 1 1 1 1 1a 2a 3a 7a 21a 21b 21c 21d 7b 21e 21f 7c 2P 1a 1a 3a 7b 21d 21c 21e 21f 7c 21a 21b 7a 3P 1a 2a 1a 7c 7c 7c 7a 7a 7a 7b 7b 7b 5P 1a 2a 3a 7b 21c 21d 21f 21e 7c 21b 21a 7a 7P 1a 2a 3a 1a 3a 3a 3a 3a 1a 3a 3a 1a 11P 1a 2a 3a 7c 21e 21f 21b 21a 7a 21c 21d 7b 13P 1a 2a 3a 7a 21b 21a 21d 21c 7b 21f 21e 7c 17P 1a 2a 3a 7c 21f 21e 21a 21b 7a 21d 21c 7b 19P 1a 2a 3a 7b 21d 21c 21e 21f 7c 21a 21b 7a 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 2 . -1 2 -1 -1 -1 -1 2 -1 -1 2 X.4 2 . 2 A A A C C C B B B X.5 2 . 2 B B B A A A C C C X.6 2 . 2 C C C B B B A A A X.7 2 . -1 C D E I H B F G A X.8 2 . -1 C E D H I B G F A X.9 2 . -1 A F G E D C I H B X.10 2 . -1 A G F D E C H I B X.11 2 . -1 B H I F G A D E C X.12 2 . -1 B I H G F A E D C A = E(7)^2+E(7)^5 B = E(7)+E(7)^6 C = E(7)^3+E(7)^4 D = E(21)^5+E(21)^16 E = E(21)^2+E(21)^19 F = E(21)^8+E(21)^13 G = E(21)+E(21)^20 H = E(21)^10+E(21)^11 I = E(21)^4+E(21)^17