Galois group: 36T563

• Label: 36T563
• Degree: $36$
• Order: $432$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_6\times S_3\times A_4$

Group invariants

Abstract group:		$C_6\times S_3\times A_4$
Order:		$432=2^{4} \cdot 3^{3}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$36$
Transitive number $t$:		$563$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$6$
Generators:		$(1,9,4,12,6,8)(2,10,3,11,5,7)(13,21,15,24,18,19)(14,22,16,23,17,20)(25,33,27,36,29,32)(26,34,28,35,30,31)$, $(1,8,30,35,17,24)(2,7,29,36,18,23)(3,9,25,32,14,20)(4,10,26,31,13,19)(5,11,28,33,15,21)(6,12,27,34,16,22)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$ x 4
$4$:	$C_2^2$
$6$:	$S_3$, $C_6$ x 12
$9$:	$C_3^2$
$12$:	$A_4$, $D_{6}$, $C_6\times C_2$ x 4
$18$:	$S_3\times C_3$ x 4, $C_6 \times C_3$ x 3
$24$:	$A_4\times C_2$ x 3
$36$:	$C_6\times S_3$ x 4, $C_3\times A_4$, 36T4
$48$:	$C_2^2 \times A_4$
$54$:	$C_3^2\times S_3$
$72$:	12T43, 18T25 x 3
$108$:	36T64
$144$:	18T60, 36T103
$216$:	24T563

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$, $S_3\times C_3$ x 3, $A_4\times C_2$ x 2

Degree 9: None

Degree 12: $C_2^2 \times A_4$

Degree 18: $C_3^2\times S_3$

Low degree siblings

There are no siblings with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Conjugacy classes not computed

Character table

72 x 72 character table

Regular extensions

Data not computed