# Properties

 Label 21T20 Degree $21$ Order $336$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $SO(3,7)$

# Related objects

## Group action invariants

 Degree $n$: $21$ Transitive number $t$: $20$ Group: $SO(3,7)$ Parity: $-1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,16,5,9,3,10,4,17)(2,19,6,8)(7,13,21,11,12,14,20,18), (1,17,2,18,13,11,19)(3,12,15,20,9,10,5)(4,8,14,6,16,7,21)

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Degree 3: None

Degree 7: None

## Low degree siblings

8T43, 14T16, 16T713, 24T707, 28T42, 28T46, 42T81, 42T82, 42T83

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $28$ $2$ $( 3,14)( 4, 7)( 6,10)( 8,20)( 9,11)(12,13)(15,16)(17,21)(18,19)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $21$ $2$ $( 2, 4)( 5, 6)( 7, 9)(10,11)(12,16)(14,20)(15,19)(17,18)$ $8, 8, 4, 1$ $42$ $8$ $( 2, 4, 9,10, 5, 6,11, 7)( 3,20, 8,14)(12,13,16,19,17,21,18,15)$ $8, 8, 4, 1$ $42$ $8$ $( 2, 6, 9, 7, 5, 4,11,10)( 3,20, 8,14)(12,21,16,15,17,13,18,19)$ $4, 4, 4, 4, 2, 2, 1$ $42$ $4$ $( 2, 9, 5,11)( 3, 8)( 4,10, 6, 7)(12,16,17,18)(13,19,21,15)(14,20)$ $6, 6, 6, 3$ $56$ $6$ $( 1, 2, 3,16, 8,10)( 4,12,17, 9, 7, 5)( 6,13,18,19,11,14)(15,21,20)$ $3, 3, 3, 3, 3, 3, 3$ $56$ $3$ $( 1, 2, 4)( 3, 6, 5)( 7,13, 9)( 8,15,19)(10,12,20)(11,14,16)(17,18,21)$ $7, 7, 7$ $48$ $7$ $( 1, 2,12,18,21,19, 9)( 3,15,17,20,10, 4, 7)( 5,13, 8, 6,14,16,11)$

## Group invariants

 Order: $336=2^{4} \cdot 3 \cdot 7$ Cyclic: no Abelian: no Solvable: no GAP id: [336, 208]
 Character table:  2 4 2 4 3 3 3 1 1 . 3 1 1 . . . . 1 1 . 7 1 . . . . . . . 1 1a 2a 2b 8a 8b 4a 6a 3a 7a 2P 1a 1a 1a 4a 4a 2b 3a 3a 7a 3P 1a 2a 2b 8b 8a 4a 2a 1a 7a 5P 1a 2a 2b 8b 8a 4a 6a 3a 7a 7P 1a 2a 2b 8a 8b 4a 6a 3a 1a X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 -1 1 -1 1 1 X.3 6 . -2 . . 2 . . -1 X.4 6 . 2 A -A . . . -1 X.5 6 . 2 -A A . . . -1 X.6 7 -1 -1 1 1 -1 -1 1 . X.7 7 1 -1 -1 -1 -1 1 1 . X.8 8 -2 . . . . 1 -1 1 X.9 8 2 . . . . -1 -1 1 A = -E(8)+E(8)^3 = -Sqrt(2) = -r2