Properties

Label 21T20
Order \(336\)
n \(21\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $SO(3,7)$

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $20$
Group :  $SO(3,7)$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

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 TypeSizeOrderRepresentative
$ 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