Galois group: 24T1868

• Label: 24T1868
• Degree: $24$
• Order: $768$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_2^3:C_4\times S_4$

Group invariants

Abstract group:		$C_2^3:C_4\times S_4$
Order:		$768=2^{8} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$24$
Transitive number $t$:		$1868$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(3,4)(5,7,6,8)(9,23,10,24)(11,22,12,21)(13,20)(14,19)(15,18)(16,17)$, $(1,7,2,8)(3,6,4,5)(9,18)(10,17)(11,19)(12,20)(13,24)(14,23)(15,22)(16,21)$, $(1,9,4,11)(2,10,3,12)(5,16,8,13)(6,15,7,14)(17,22,20,23)(18,21,19,24)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_4$ x 4, $C_2^2$ x 7
$6$:	$S_3$
$8$:	$D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$12$:	$D_{6}$ x 3
$16$:	$D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$24$:	$S_4$, $S_3 \times C_2^2$, $S_3 \times C_4$ x 2
$32$:	$C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$
$48$:	$S_4\times C_2$ x 3, 12T28 x 2, 24T27
$64$:	16T76
$96$:	12T48, 12T53 x 2, 24T146
$192$:	12T86 x 2, 24T337, 24T397
$384$:	24T1070

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Degree 8: None

Degree 12: 12T28

Low degree siblings

24T1719 x 2, 24T1721 x 2, 24T1868 x 3, 32T35066 x 4, 32T35073 x 2, 32T35194 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Conjugacy classes not computed

Character table

55 x 55 character table

Regular extensions

Data not computed