# Properties

 Label 21T12 Degree $21$ Order $147$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_7:(C_7:C_3)$

# Related objects

## Group action invariants

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

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 7: None

## Low degree siblings

21T12

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

## Group invariants

 Order: $147=3 \cdot 7^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [147, 5]
 Character table:  3 1 . . . . . . . . . . . . . . . . 1 1 7 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 . . 1a 7a 7b 7c 7d 7e 7f 7g 7h 7i 7j 7k 7l 7m 7n 7o 7p 3a 3b 2P 1a 7b 7d 7f 7a 7c 7e 7k 7h 7l 7j 7o 7p 7m 7n 7g 7i 3b 3a 3P 1a 7c 7f 7b 7e 7a 7d 7i 7n 7k 7m 7l 7o 7j 7h 7p 7g 1a 1a 5P 1a 7e 7c 7a 7f 7d 7b 7p 7n 7g 7m 7i 7k 7j 7h 7l 7o 3b 3a 7P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 3a 3b X.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 H /H X.3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 /H H X.4 3 A A /A A /A /A /A 3 A A /A A /A 3 /A A . . X.5 3 /A /A A /A A A A 3 /A /A A /A A 3 A /A . . X.6 3 A A /A A /A /A A /A /A 3 A /A 3 A A /A . . X.7 3 /A /A A /A A A /A A A 3 /A A 3 /A /A A . . X.8 3 B D C C D B E /A /G /A F /E A A G /F . . X.9 3 C B D D B C G /A /F /A E /G A A F /E . . X.10 3 D C B B C D F /A /E /A G /F A A E /G . . X.11 3 B D C C D B /E A G A /F E /A /A /G F . . X.12 3 C B D D B C /G A F A /E G /A /A /F E . . X.13 3 D C B B C D /F A E A /G F /A /A /E G . . X.14 3 E F /G G /F /E C /A D A B C /A A D B . . X.15 3 F G /E E /G /F B /A C A D B /A A C D . . X.16 3 G E /F F /E /G D /A B A C D /A A B C . . X.17 3 /F /G E /E G F B A C /A D B A /A C D . . X.18 3 /E /F G /G F E C A D /A B C A /A D B . . X.19 3 /G /E F /F E G D A B /A C D A /A B C . . A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7 B = -E(7)^2-E(7)^3-E(7)^4-E(7)^5 C = -E(7)-E(7)^2-E(7)^5-E(7)^6 D = -E(7)-E(7)^3-E(7)^4-E(7)^6 E = 2*E(7)+E(7)^5 F = 2*E(7)^2+E(7)^3 G = 2*E(7)^4+E(7)^6 H = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3