# Properties

 Label 21T11 Degree $21$ Order $126$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3\times C_7:C_3$

# Related objects

## Group action invariants

 Degree $n$: $21$ Transitive number $t$: $11$ Group: $S_3\times C_7:C_3$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,12,5)(2,10,6)(3,11,4)(7,15,17)(8,13,18)(9,14,16)(19,21,20), (1,6,16,3,4,18)(2,5,17)(7,9)(10,21,13,12,19,15)(11,20,14)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$21$:  $C_7:C_3$
$42$:  $(C_7:C_3) \times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

Degree 7: $C_7:C_3$

## Low degree siblings

42T19, 42T23

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

## Group invariants

 Order: $126=2 \cdot 3^{2} \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [126, 8]
 Character table:  2 1 1 1 1 1 1 . . . 1 1 . 1 1 . 3 2 2 2 1 1 1 2 2 2 1 . 1 1 . 1 7 1 . . 1 . . 1 . . 1 1 1 1 1 1 1a 3a 3b 2a 6a 6b 3c 3d 3e 7a 14a 21a 7b 14b 21b 2P 1a 3b 3a 1a 3b 3a 3c 3e 3d 7a 7a 21a 7b 7b 21b 3P 1a 1a 1a 2a 2a 2a 1a 1a 1a 7b 14b 7b 7a 14a 7a 5P 1a 3b 3a 2a 6b 6a 3c 3e 3d 7b 14b 21b 7a 14a 21a 7P 1a 3a 3b 2a 6a 6b 3c 3d 3e 1a 2a 3c 1a 2a 3c 11P 1a 3b 3a 2a 6b 6a 3c 3e 3d 7a 14a 21a 7b 14b 21b 13P 1a 3a 3b 2a 6a 6b 3c 3d 3e 7b 14b 21b 7a 14a 21a 17P 1a 3b 3a 2a 6b 6a 3c 3e 3d 7b 14b 21b 7a 14a 21a 19P 1a 3a 3b 2a 6a 6b 3c 3d 3e 7b 14b 21b 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 A /A -1 -A -/A 1 A /A 1 -1 1 1 -1 1 X.4 1 /A A -1 -/A -A 1 /A A 1 -1 1 1 -1 1 X.5 1 A /A 1 A /A 1 A /A 1 1 1 1 1 1 X.6 1 /A A 1 /A A 1 /A A 1 1 1 1 1 1 X.7 2 2 2 . . . -1 -1 -1 2 . -1 2 . -1 X.8 2 B /B . . . -1 -A -/A 2 . -1 2 . -1 X.9 2 /B B . . . -1 -/A -A 2 . -1 2 . -1 X.10 3 . . -3 . . 3 . . C -C C /C -/C /C X.11 3 . . -3 . . 3 . . /C -/C /C C -C C X.12 3 . . 3 . . 3 . . C C C /C /C /C X.13 3 . . 3 . . 3 . . /C /C /C C C C X.14 6 . . . . . -3 . . D . -C /D . -/C X.15 6 . . . . . -3 . . /D . -/C D . -C A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 C = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 D = 2*E(7)^3+2*E(7)^5+2*E(7)^6 = -1-Sqrt(-7) = -1-i7