Show commands:
Magma
magma: G := TransitiveGroup(32, 1016);
Group action invariants
| Degree $n$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1016$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2\times D_4^2$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $8$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,7,31,11)(2,8,32,12)(3,5,29,9)(4,6,30,10)(13,21,19,28)(14,22,20,27)(15,23,17,26)(16,24,18,25), (1,32)(2,31)(3,30)(4,29)(5,6)(7,8)(9,10)(11,12)(21,28)(22,27)(23,26)(24,25), (1,4)(2,3)(5,12)(6,11)(7,10)(8,9)(13,16)(14,15)(17,20)(18,19)(21,25)(22,26)(23,27)(24,28)(29,32)(30,31), (1,16)(2,15)(3,14)(4,13)(5,21)(6,22)(7,23)(8,24)(9,28)(10,27)(11,26)(12,25)(17,32)(18,31)(19,30)(20,29), (1,15)(2,16)(3,13)(4,14)(5,27)(6,28)(7,25)(8,26)(9,22)(10,21)(11,24)(12,23)(17,31)(18,32)(19,29)(20,30) | magma: Generators(G);
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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 16, $C_2^3$ x 155 $16$: $D_4\times C_2$ x 56, $C_2^4$ x 31 $32$: $Q_8:C_2^2$ x 2, $C_2^2 \times D_4$ x 28, 32T39 $64$: 16T69, 16T109 x 4, 32T273 x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 4: $C_2^2$ x 7, $D_{4}$ x 8
Degree 8: $C_2^3$, $D_4\times C_2$ x 12, $Q_8:C_2^2$ x 2
Degree 16: $C_2^2 \times D_4$ x 2, 16T69, 16T109 x 4
Low degree siblings
32T1016 x 63Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 50 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $128=2^{7}$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $2$ | ||
| Label: | 128.2194 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);