Properties

Label 45T10923
Degree $45$
Order $1.196\times 10^{56}$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_{45}$

Related objects

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Group action invariants

Degree $n$:  $45$
Transitive number $t$:  $10923$
Group:  $S_{45}$
Parity:  $-1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,2), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 3: None

Degree 5: None

Degree 9: None

Degree 15: None

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 89,134 conjugacy classes of elements. Data not shown.

Group invariants

Order:  $119622220865480194561963161495657715064383733760000000000=2^{41} \cdot 3^{21} \cdot 5^{10} \cdot 7^{6} \cdot 11^{4} \cdot 13^{3} \cdot 17^{2} \cdot 19^{2} \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 41 \cdot 43$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  not available
Character table: not available.