Properties

Label 45T39
Degree $45$
Order $270$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times C_3^2:S_3$

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Group action invariants

Degree $n$:  $45$
Transitive number $t$:  $39$
Group:  $C_5\times C_3^2:S_3$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $15$
Generators:  (1,41,21,13,38,32,11,5,28,23,2,40,19,15,37,31,12,4,29,24,3,42,20,14,39,33,10,6,30,22)(7,17,26,34,45,9,18,27,35,43,8,16,25,36,44), (4,5,6)(7,8,9)(13,14,15)(16,18,17)(22,24,23)(25,27,26)(31,32,33)(34,36,35)(40,41,42)(43,45,44), (1,2,3)(4,34,5,35,6,36)(7,24,9,23,8,22)(10,11,12)(13,45,14,43,15,44)(16,31,17,32,18,33)(19,20,21)(25,41,26,42,27,40)(28,29,30)(37,39,38)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$ x 4
$10$:  $C_{10}$
$18$:  $C_3^2:C_2$
$54$:  $(C_3^2:C_3):C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $C_5$

Degree 9: $(C_3^2:C_3):C_2$

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, 17]
Character table: not available.