Properties

Label 45T144
Degree $45$
Order $1080$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no

Related objects

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Group action invariants

Degree $n$:  $45$
Transitive number $t$:  $144$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,22,10,5)(2,24,12,6)(3,23,11,4)(7,26,34,17)(8,27,36,16)(9,25,35,18)(13,28)(14,30)(15,29)(19,31,37,41)(20,32,39,40)(21,33,38,42)(43,44), (1,42,44,20,32,26,29,5,16,10,15,36,3,40,45,21,31,27,30,6,18,11,13,34,2,41,43,19,33,25,28,4,17,12,14,35)(7,38,22,9,37,24,8,39,23)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$18$:  $S_3\times C_3$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$
$54$:  $(C_9:C_3):C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: $S_3$

Degree 5: $F_5$

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

Degree 15: $F_5 \times S_3$

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:  $1080=2^{3} \cdot 3^{3} \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [1080, 271]
Character table: not available.