# Properties

 Label 21T7 Degree $21$ Order $63$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3\times C_7:C_3$

# Related objects

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$21$:  $C_7:C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 7: $C_7:C_3$

## Low degree siblings

21T7 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 1, 1, 1$ $7$ $3$ $( 4, 9,14)( 5, 7,15)( 6, 8,13)(10,20,18)(11,21,16)(12,19,17)$ $3, 3, 3, 3, 3, 3, 1, 1, 1$ $7$ $3$ $( 4,14, 9)( 5,15, 7)( 6,13, 8)(10,18,20)(11,16,21)(12,17,19)$ $3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$ $3, 3, 3, 3, 3, 3, 3$ $7$ $3$ $( 1, 2, 3)( 4, 7,13)( 5, 8,14)( 6, 9,15)(10,21,17)(11,19,18)(12,20,16)$ $3, 3, 3, 3, 3, 3, 3$ $7$ $3$ $( 1, 2, 3)( 4,15, 8)( 5,13, 9)( 6,14, 7)(10,16,19)(11,17,20)(12,18,21)$ $3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$ $3, 3, 3, 3, 3, 3, 3$ $7$ $3$ $( 1, 3, 2)( 4, 8,15)( 5, 9,13)( 6, 7,14)(10,19,16)(11,20,17)(12,21,18)$ $3, 3, 3, 3, 3, 3, 3$ $7$ $3$ $( 1, 3, 2)( 4,13, 7)( 5,14, 8)( 6,15, 9)(10,17,21)(11,18,19)(12,16,20)$ $7, 7, 7$ $3$ $7$ $( 1, 4, 9,10,14,18,20)( 2, 5, 7,11,15,16,21)( 3, 6, 8,12,13,17,19)$ $21$ $3$ $21$ $( 1, 5, 8,10,15,17,20, 2, 6, 9,11,13,18,21, 3, 4, 7,12,14,16,19)$ $21$ $3$ $21$ $( 1, 6, 7,10,13,16,20, 3, 5, 9,12,15,18,19, 2, 4, 8,11,14,17,21)$ $7, 7, 7$ $3$ $7$ $( 1,10,20, 9,18, 4,14)( 2,11,21, 7,16, 5,15)( 3,12,19, 8,17, 6,13)$ $21$ $3$ $21$ $( 1,11,19, 9,16, 6,14, 2,12,20, 7,17, 4,15, 3,10,21, 8,18, 5,13)$ $21$ $3$ $21$ $( 1,12,21, 9,17, 5,14, 3,11,20, 8,16, 4,13, 2,10,19, 7,18, 6,15)$

## Group invariants

 Order: $63=3^{2} \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [63, 3]
 Character table:  3 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 7 1 . . 1 . . 1 . . 1 1 1 1 1 1 1a 3a 3b 3c 3d 3e 3f 3g 3h 7a 21a 21b 7b 21c 21d 2P 1a 3b 3a 3f 3h 3g 3c 3e 3d 7a 21b 21a 7b 21d 21c 3P 1a 1a 1a 1a 1a 1a 1a 1a 1a 7b 7b 7b 7a 7a 7a 5P 1a 3b 3a 3f 3h 3g 3c 3e 3d 7b 21d 21c 7a 21b 21a 7P 1a 3a 3b 3c 3d 3e 3f 3g 3h 1a 3c 3f 1a 3c 3f 11P 1a 3b 3a 3f 3h 3g 3c 3e 3d 7a 21b 21a 7b 21d 21c 13P 1a 3a 3b 3c 3d 3e 3f 3g 3h 7b 21c 21d 7a 21a 21b 17P 1a 3b 3a 3f 3h 3g 3c 3e 3d 7b 21d 21c 7a 21b 21a 19P 1a 3a 3b 3c 3d 3e 3f 3g 3h 7b 21c 21d 7a 21a 21b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 A A A /A /A /A 1 A /A 1 A /A X.3 1 1 1 /A /A /A A A A 1 /A A 1 /A A 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 A /A 1 /A 1 A 1 A /A 1 A /A X.7 1 /A A /A A 1 A 1 /A 1 /A A 1 /A A X.8 1 A /A /A 1 A A /A 1 1 /A A 1 /A A X.9 1 /A A A 1 /A /A A 1 1 A /A 1 A /A 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 . . B . . /B . . C D E /C /E /D X.13 3 . . /B . . B . . C E D /C /D /E X.14 3 . . B . . /B . . /C /E /D C D E X.15 3 . . /B . . B . . /C /D /E C E D A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 C = E(7)+E(7)^2+E(7)^4 = (-1+Sqrt(-7))/2 = b7 D = E(21)^5+E(21)^17+E(21)^20 E = E(21)^10+E(21)^13+E(21)^19