# Properties

 Label 15T44 Degree $15$ Order $2430$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no

# Related objects

## Group action invariants

 Degree $n$: $15$ Transitive number $t$: $44$ CHM label: $[3^{5}:2]5$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $\card{\Aut(F/K)}$: $1$ Generators: (1,11)(2,7)(4,14)(5,10)(8,13), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15), (5,10,15)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$
$30$:  $S_3 \times C_5$
$810$:  15T33

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: None

Degree 5: $C_5$

## Low degree siblings

15T44 x 15, 30T394 x 16, 45T253 x 8, 45T254 x 16, 45T255 x 32, 45T256 x 32, 45T257 x 64, 45T258 x 64

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 5,15,10)$ $3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 3,13, 8)( 5,15,10)$ $3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 3,13, 8)( 5,10,15)$ $3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 1,11, 6)( 5,15,10)$ $3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 1,11, 6)( 5,10,15)$ $3, 3, 3, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 1,11, 6)( 3,13, 8)( 5,15,10)$ $3, 3, 3, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 1,11, 6)( 3,13, 8)( 5,10,15)$ $3, 3, 3, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 1,11, 6)( 3, 8,13)( 5,15,10)$ $3, 3, 3, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 1,11, 6)( 3, 8,13)( 5,10,15)$ $3, 3, 3, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 3,13, 8)( 4,14, 9)( 5,15,10)$ $3, 3, 3, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 3,13, 8)( 4,14, 9)( 5,10,15)$ $3, 3, 3, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 3, 8,13)( 4,14, 9)( 5,15,10)$ $3, 3, 3, 1, 1, 1, 1, 1, 1$ $10$ $3$ $( 3, 8,13)( 4,14, 9)( 5,10,15)$ $3, 3, 3, 3, 1, 1, 1$ $10$ $3$ $( 1,11, 6)( 3,13, 8)( 4,14, 9)( 5,15,10)$ $3, 3, 3, 3, 1, 1, 1$ $10$ $3$ $( 1,11, 6)( 3,13, 8)( 4,14, 9)( 5,10,15)$ $3, 3, 3, 3, 1, 1, 1$ $10$ $3$ $( 1,11, 6)( 3, 8,13)( 4,14, 9)( 5,15,10)$ $3, 3, 3, 3, 1, 1, 1$ $10$ $3$ $( 1,11, 6)( 3, 8,13)( 4,14, 9)( 5,10,15)$ $3, 3, 3, 3, 1, 1, 1$ $10$ $3$ $( 1, 6,11)( 3,13, 8)( 4,14, 9)( 5,15,10)$ $3, 3, 3, 3, 1, 1, 1$ $10$ $3$ $( 1, 6,11)( 3,13, 8)( 4,14, 9)( 5,10,15)$ $3, 3, 3, 3, 1, 1, 1$ $10$ $3$ $( 1, 6,11)( 3, 8,13)( 4,14, 9)( 5,15,10)$ $3, 3, 3, 3, 1, 1, 1$ $10$ $3$ $( 1, 6,11)( 3, 8,13)( 4,14, 9)( 5,10,15)$ $3, 3, 3, 3, 3$ $2$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$ $3, 3, 3, 3, 3$ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,10,15)$ $3, 3, 3, 3, 3$ $10$ $3$ $( 1,11, 6)( 2,12, 7)( 3, 8,13)( 4,14, 9)( 5,10,15)$ $3, 3, 3, 3, 3$ $10$ $3$ $( 1, 6,11)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,10,15)$ $2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $243$ $2$ $( 6,11)( 7,12)( 8,13)( 9,14)(10,15)$ $5, 5, 5$ $81$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ $15$ $162$ $15$ $( 1, 4, 7, 5, 8,11,14, 2,15, 3, 6, 9,12,10,13)$ $10, 5$ $243$ $10$ $( 1, 4,12,10, 8, 6,14, 2, 5,13)( 3,11, 9, 7,15)$ $5, 5, 5$ $81$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ $15$ $162$ $15$ $( 1, 7,13, 4, 5,11, 2, 8,14,15, 6,12, 3, 9,10)$ $10, 5$ $243$ $10$ $( 1,12, 3,14, 5, 6, 7, 8, 9,10)( 2,13, 4,15,11)$ $5, 5, 5$ $81$ $5$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$ $15$ $162$ $15$ $( 1,13, 5, 2,14,11, 8,15,12, 9, 6, 3,10, 7, 4)$ $10, 5$ $243$ $10$ $( 1, 8, 5, 2, 9,11,13,15, 7, 4)( 3,10,12,14, 6)$ $5, 5, 5$ $81$ $5$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$ $15$ $162$ $15$ $( 1, 5,14, 8, 2,11,15, 9, 3,12, 6,10, 4,13, 7)$ $10, 5$ $243$ $10$ $( 1,15,14,13,12,11, 5, 9, 3, 7)( 2, 6,10, 4, 8)$

## Group invariants

 Order: $2430=2 \cdot 3^{5} \cdot 5$ Cyclic: no Abelian: no Solvable: yes Label: not available
 Character table: not available.