Properties

Label 45T38
Degree $45$
Order $270$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times D_9:C_3$

Learn more about

Group action invariants

Degree $n$:  $45$
Transitive number $t$:  $38$
Group:  $C_5\times D_9:C_3$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $5$
Generators:  (1,25,21,45,38,17,11,34,29,7,2,26,19,43,37,18,12,35,30,9,3,27,20,44,39,16,10,36,28,8)(4,15,24,31,42,5,14,23,33,40)(6,13,22,32,41), (1,38,30,20,11,2,39,28,19,12)(3,37,29,21,10)(4,25,31,7,13,35,42,16,23,45,6,26,33,9,15,36,41,17,24,43,5,27,32,8,14,34,40,18,22,44)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$5$:  $C_5$
$6$:  $S_3$, $C_6$
$10$:  $C_{10}$
$18$:  $S_3\times C_3$
$54$:  $(C_9:C_3):C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $C_5$

Degree 9: $(C_9:C_3):C_2$

Degree 15: $S_3 \times C_5$

Low degree siblings

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

Conjugacy classes

There are 50 conjugacy classes of elements. Data not shown.

Group invariants

Order:  $270=2 \cdot 3^{3} \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [270, 11]
Character table: not available.