# Properties

 Label 32T12882 Degree $32$ Order $512$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $D_4^2:C_2^3$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(32, 12882);

## Group action invariants

 Degree $n$: $32$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $12882$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $D_4^2:C_2^3$ 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: (1,28,17,10)(2,27,18,9)(3,26,19,12)(4,25,20,11)(5,6)(7,8)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32), (1,21,27,29,17,7,9,15)(2,22,28,30,18,8,10,16)(3,23,25,31,19,5,11,13)(4,24,26,32,20,6,12,14), (1,14,9,24,17,32,27,6)(2,13,10,23,18,31,28,5)(3,16,11,22,19,30,25,8)(4,15,12,21,20,29,26,7), (1,32)(2,31)(3,30)(4,29)(5,10,23,28)(6,9,24,27)(7,12,21,26)(8,11,22,25)(13,18)(14,17)(15,20)(16,19), (1,16,27,8,17,30,9,22)(2,15,28,7,18,29,10,21)(3,14,25,6,19,32,11,24)(4,13,26,5,20,31,12,23) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 31
$4$:  $C_2^2$ x 155
$8$:  $D_{4}$ x 24, $C_2^3$ x 155
$16$:  $D_4\times C_2$ x 84, $C_2^4$ x 31
$32$:  $C_2^2 \wr C_2$ x 16, $C_2^2 \times D_4$ x 42, 32T39
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 4, 16T105 x 12, 32T273 x 3
$128$:  $C_2 \wr C_2\wr C_2$ x 4, 16T245 x 6, 32T1369
$256$:  16T509 x 6, 32T4287

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4\times C_2$ x 6, $C_2 \wr C_2\wr C_2$ x 4

Degree 16: $C_2^2 \times D_4$, 16T509 x 6

## Low degree siblings

32T12882 x 127, 32T12885 x 384

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 $a(G)$:     $1/4$

## Group invariants

 Order: $512=2^{9}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $4$ Label: 512.7530050 magma: IdentifyGroup(G); Character table: 80 x 80 character table

magma: CharacterTable(G);