# Properties

 Label 32T34907 Degree $32$ Order $768$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^5:S_4$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(32, 34907);

## Group action invariants

 Degree $n$: $32$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $34907$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_2^5:S_4$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $8$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,8,23,20)(2,7,24,19)(3,6,21,18)(4,5,22,17)(9,30,15,28)(10,29,16,27)(11,32,13,26)(12,31,14,25), (1,18,15,24,5,10)(2,17,16,23,6,9)(3,20,13,22,7,12)(4,19,14,21,8,11)(25,32)(26,31)(27,30)(28,29), (1,12,23,14)(2,11,24,13)(3,10,21,16)(4,9,22,15)(5,20)(6,19)(7,18)(8,17)(25,28)(26,27)(29,32)(30,31) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$24$:  $S_4$ x 3, $S_3 \times C_2^2$
$48$:  $S_4\times C_2$ x 9
$96$:  $V_4^2:S_3$, 12T48 x 3
$192$:  $C_2^3:S_4$ x 4, 12T100 x 3
$384$:  12T139, 16T747 x 6

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $S_4$

Degree 8: $S_4\times C_2$ x 3, $C_2^3:S_4$ x 4

Degree 16: 16T182, 16T747 x 6

## Low degree siblings

32T34907 x 23

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/8$

## Group invariants

 Order: $768=2^{8} \cdot 3$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 768.1090213 magma: IdentifyGroup(G); Character table: 52 x 52 character table

magma: CharacterTable(G);