Properties

Label 45T37
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 $t$ :  $37$
Group :  $C_5\times He_3:C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,44,12,7,21,16,30,26,38,36,3,43,11,8,20,18,29,25,39,35,2,45,10,9,19,17,28,27,37,34)(4,31,14,40,24,5,33,15,42,23)(6,32,13,41,22), (1,3)(4,36)(5,35)(6,34)(7,24)(8,23)(9,22)(10,11)(13,43)(14,45)(15,44)(16,32)(17,31)(18,33)(19,21)(25,41)(26,40)(27,42)(29,30)(37,39)
$|\Aut(F/K)|$:  $5$

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: $S_3$

Degree 5: $C_5$

Degree 9: $C_3^2 : C_6$

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, 10]
Character table: Data not available.