# Properties

 Label 21T18 Degree $21$ Order $294$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_7^2:S_3$

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## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

Degree 7: None

## Low degree siblings

14T15, 21T17, 42T56, 42T57, 42T62

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $7, 7, 1, 1, 1, 1, 1, 1, 1$ $6$ $7$ $( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$ $7, 7, 1, 1, 1, 1, 1, 1, 1$ $6$ $7$ $( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$ $7, 7, 1, 1, 1, 1, 1, 1, 1$ $6$ $7$ $( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$ $2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1$ $21$ $2$ $( 8,15)( 9,21)(10,20)(11,19)(12,18)(13,17)(14,16)$ $7, 7, 7$ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$ $7, 7, 7$ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,18,21,17,20,16,19)$ $7, 7, 7$ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$ $14, 7$ $21$ $14$ $( 1, 2, 3, 4, 5, 6, 7)( 8,15,14,16,13,17,12,18,11,19,10,20, 9,21)$ $7, 7, 7$ $3$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$ $7, 7, 7$ $3$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$ $14, 7$ $21$ $14$ $( 1, 3, 5, 7, 2, 4, 6)( 8,15,13,17,11,19, 9,21,14,16,12,18,10,20)$ $7, 7, 7$ $6$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$ $14, 7$ $21$ $14$ $( 1, 4, 7, 3, 6, 2, 5)( 8,15,12,18, 9,21,13,17,10,20,14,16,11,19)$ $7, 7, 7$ $3$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$ $7, 7, 7$ $3$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$ $14, 7$ $21$ $14$ $( 1, 5, 2, 6, 3, 7, 4)( 8,15,11,19,14,16,10,20,13,17, 9,21,12,18)$ $14, 7$ $21$ $14$ $( 1, 6, 4, 2, 7, 5, 3)( 8,15,10,20,12,18,14,16, 9,21,11,19,13,17)$ $14, 7$ $21$ $14$ $( 1, 7, 6, 5, 4, 3, 2)( 8,15, 9,21,10,20,11,19,12,18,13,17,14,16)$ $3, 3, 3, 3, 3, 3, 3$ $98$ $3$ $( 1, 8,15)( 2, 9,21)( 3,10,20)( 4,11,19)( 5,12,18)( 6,13,17)( 7,14,16)$

## Group invariants

 Order: $294=2 \cdot 3 \cdot 7^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [294, 7]
 Character table:  2 1 . . . 1 1 . 1 1 1 1 1 . 1 1 1 1 1 1 . 3 1 . . . . . . . . . . . . . . . . . . 1 7 2 2 2 2 1 2 2 2 1 2 2 1 2 1 2 2 1 1 1 . 1a 7a 7b 7c 2a 7d 7e 7f 14a 7g 7h 14b 7i 14c 7j 7k 14d 14e 14f 3a 2P 1a 7b 7c 7a 1a 7g 7e 7h 7h 7j 7k 7k 7i 7j 7d 7f 7f 7g 7d 3a 3P 1a 7c 7a 7b 2a 7f 7i 7g 14c 7h 7j 14f 7e 14b 7k 7d 14e 14a 14d 1a 5P 1a 7b 7c 7a 2a 7k 7i 7d 14e 7f 7g 14c 7e 14a 7h 7j 14f 14d 14b 3a 7P 1a 1a 1a 1a 2a 1a 1a 1a 2a 1a 1a 2a 1a 2a 1a 1a 2a 2a 2a 3a 11P 1a 7c 7a 7b 2a 7j 7e 7k 14d 7d 7f 14a 7i 14e 7g 7h 14b 14f 14c 3a 13P 1a 7a 7b 7c 2a 7h 7i 7j 14f 7k 7d 14e 7e 14d 7f 7g 14c 14b 14a 3a X.1 1 1 1 1 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 1 -1 -1 -1 1 X.3 2 2 2 2 . 2 2 2 . 2 2 . 2 . 2 2 . . . -1 X.4 3 A C B -1 G N H P /I /G /R /N Q /H I /Q R /P . X.5 3 A C B -1 /G /N /H /P I G R N /Q H /I Q /R P . X.6 3 B A C -1 H /N /I Q /G /H /P N /R I G R P /Q . X.7 3 B A C -1 /H N I /Q G H P /N R /I /G /R /P Q . X.8 3 C B A -1 I /N G R H /I Q N P /G /H /P /Q /R . X.9 3 C B A -1 /I N /G /R /H I /Q /N /P G H P Q R . X.10 3 A C B 1 G N H -P /I /G -/R /N -Q /H I -/Q -R -/P . X.11 3 A C B 1 /G /N /H -/P I G -R N -/Q H /I -Q -/R -P . X.12 3 B A C 1 H /N /I -Q /G /H -/P N -/R I G -R -P -/Q . X.13 3 B A C 1 /H N I -/Q G H -P /N -R /I /G -/R -/P -Q . X.14 3 C B A 1 I /N G -R H /I -Q N -P /G /H -/P -/Q -/R . X.15 3 C B A 1 /I N /G -/R /H I -/Q /N -/P G H -P -Q -R . X.16 6 D E F . J -1 L . K J . -1 . L K . . . . X.17 6 E F D . K -1 J . L K . -1 . J L . . . . X.18 6 F D E . L -1 K . J L . -1 . K J . . . . X.19 6 -1 -1 -1 . M O /M . M /M . /O . M /M . . . . X.20 6 -1 -1 -1 . /M /O M . /M M . O . /M M . . . . A = -E(7)-E(7)^2-E(7)^5-E(7)^6 B = -E(7)-E(7)^3-E(7)^4-E(7)^6 C = -E(7)^2-E(7)^3-E(7)^4-E(7)^5 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 = 2*E(7)^4+E(7)^6 H = E(7)^4+2*E(7)^5 I = E(7)^2+2*E(7)^6 J = -2*E(7)-2*E(7)^2-2*E(7)^5-2*E(7)^6 K = -2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5 L = -2*E(7)-2*E(7)^3-2*E(7)^4-2*E(7)^6 M = 2*E(7)^3+2*E(7)^5+2*E(7)^6 = -1-Sqrt(-7) = -1-i7 N = E(7)+E(7)^2+E(7)^4 = (-1+Sqrt(-7))/2 = b7 O = -2*E(7)-2*E(7)^2-3*E(7)^3-2*E(7)^4-3*E(7)^5-3*E(7)^6 = (5+Sqrt(-7))/2 = 3+b7 P = -E(7)^4 Q = -E(7)^5 R = -E(7)^6