# Properties

 Label 14T16 Order $$336$$ n $$14$$ Cyclic No Abelian No Solvable No Primitive No $p$-group No Group: $SO(3,7)$

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## Group action invariants

 Degree $n$ : $14$ Transitive number $t$ : $16$ Group : $SO(3,7)$ CHM label : $L_{7}:2(14)=[L(7)_%]2$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,13,11,9,7,5,3)(2,4,6,8,10,12,14), (1,9,11)(2,4,8)(3,13,5)(6,12,10), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (2,4)(5,13)(6,12)(9,11) $|\Aut(F/K)|$: $1$

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 7: None

## Low degree siblings

8T43, 16T713, 21T20, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $21$ $2$ $( 5,11)( 6,12)( 9,13)(10,14)$ $3, 3, 3, 3, 1, 1$ $56$ $3$ $( 3, 5,11)( 4,12, 6)( 7,13, 9)( 8,10,14)$ $4, 4, 2, 2, 1, 1$ $42$ $4$ $( 2, 4,12, 6)( 3, 9, 7,11)( 5,13)( 8,14)$ $2, 2, 2, 2, 2, 2, 2$ $28$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$ $6, 6, 2$ $56$ $6$ $( 1, 2)( 3, 6, 5, 4,11,12)( 7,14,13, 8, 9,10)$ $8, 4, 2$ $42$ $8$ $( 1, 2, 3, 4)( 5,10,13,12,11,14, 9, 6)( 7, 8)$ $8, 4, 2$ $42$ $8$ $( 1, 2, 3, 4)( 5,14,13, 6,11,10, 9,12)( 7, 8)$ $7, 7$ $48$ $7$ $( 1, 3, 9,11,13, 7, 5)( 2, 4,10, 8, 6,14,12)$

## Group invariants

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