Properties

 Label 21T19 Degree $21$ Order $294$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_7:D_7:C_3$

Related objects

Group action invariants

 Degree $n$: $21$ Transitive number $t$: $19$ Group: $C_7:D_7:C_3$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,7)(2,6)(3,5)(8,13)(9,12)(10,11)(15,18)(16,17)(19,21), (1,14,21,6,8,17)(2,10,16,5,12,15)(3,13,18,4,9,20)(7,11,19)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$42$:  $F_7$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T19, 42T58 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $7, 7, 1, 1, 1, 1, 1, 1, 1$ $6$ $7$ $( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$ $7, 7, 1, 1, 1, 1, 1, 1, 1$ $6$ $7$ $( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$ $7, 7, 1, 1, 1, 1, 1, 1, 1$ $6$ $7$ $( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $49$ $2$ $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$ $7, 7, 7$ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$ $7, 7, 7$ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$ $7, 7, 7$ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$ $7, 7, 7$ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$ $7, 7, 7$ $6$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$ $6, 6, 6, 3$ $49$ $6$ $( 1, 8,15)( 2,11,17, 7,12,20)( 3,14,19, 6, 9,18)( 4,10,21, 5,13,16)$ $3, 3, 3, 3, 3, 3, 3$ $49$ $3$ $( 1, 8,15)( 2,12,17)( 3, 9,19)( 4,13,21)( 5,10,16)( 6,14,18)( 7,11,20)$ $3, 3, 3, 3, 3, 3, 3$ $49$ $3$ $( 1,15, 8)( 2,17,12)( 3,19, 9)( 4,21,13)( 5,16,10)( 6,18,14)( 7,20,11)$ $6, 6, 6, 3$ $49$ $6$ $( 1,15, 8)( 2,20,12, 7,17,11)( 3,18, 9, 6,19,14)( 4,16,13, 5,21,10)$

Group invariants

 Order: $294=2 \cdot 3 \cdot 7^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [294, 14]
 Character table:  2 1 . . . 1 . . . . . 1 1 1 1 3 1 . . . 1 . . . . . 1 1 1 1 7 2 2 2 2 . 2 2 2 2 2 . . . . 1a 7a 7b 7c 2a 7d 7e 7f 7g 7h 6a 3a 3b 6b 2P 1a 7b 7c 7a 1a 7d 7e 7h 7f 7g 3b 3b 3a 3a 3P 1a 7c 7a 7b 2a 7d 7e 7g 7h 7f 2a 1a 1a 2a 5P 1a 7b 7c 7a 2a 7d 7e 7h 7f 7g 6b 3b 3a 6a 7P 1a 1a 1a 1a 2a 1a 1a 1a 1a 1a 6a 3a 3b 6b X.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 X.3 1 1 1 1 -1 1 1 1 1 1 G -G -/G /G X.4 1 1 1 1 -1 1 1 1 1 1 /G -/G -G G X.5 1 1 1 1 1 1 1 1 1 1 -/G -/G -G -G X.6 1 1 1 1 1 1 1 1 1 1 -G -G -/G -/G X.7 6 -1 -1 -1 . 6 -1 -1 -1 -1 . . . . X.8 6 -1 -1 -1 . -1 6 -1 -1 -1 . . . . X.9 6 A B C . -1 -1 E D F . . . . X.10 6 B C A . -1 -1 F E D . . . . X.11 6 C A B . -1 -1 D F E . . . . X.12 6 D E F . -1 -1 C B A . . . . X.13 6 E F D . -1 -1 A C B . . . . X.14 6 F D E . -1 -1 B A C . . . . A = -2*E(7)-2*E(7)^2-2*E(7)^5-2*E(7)^6 B = -2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5 C = -2*E(7)-2*E(7)^3-2*E(7)^4-2*E(7)^6 D = 2*E(7)+E(7)^2+E(7)^5+2*E(7)^6 E = 2*E(7)^2+E(7)^3+E(7)^4+2*E(7)^5 F = E(7)+2*E(7)^3+2*E(7)^4+E(7)^6 G = -E(3) = (1-Sqrt(-3))/2 = -b3