Properties

Label 45T30
Order \(270\)
n \(45\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5\times He_3:C_2$

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Group action invariants

Degree $n$ :  $45$
Transitive number $t$ :  $30$
Group :  $C_5\times He_3:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,14,19,27,31,39,45,6,11,16,24,30,35,41)(2,9,13,21,26,33,37,44,5,10,17,23,29,34,40)(3,7,15,20,25,32,38,43,4,12,18,22,28,36,42), (1,8,13,19,27,33,39,45,5,11,16,23,30,35,40)(2,7,15,20,26,31,37,43,4,12,17,24,29,36,42,3,9,14,21,25,32,38,44,6,10,18,22,28,34,41)
$|\Aut(F/K)|$:  $5$

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:  $C_3^2 : C_6$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $C_3$

Degree 5: $C_5$

Degree 9: $C_3^2 : S_3 $

Degree 15: $C_{15}$

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, 10]
Character table: Data not available.