Show commands:
Magma
magma: G := TransitiveGroup(32, 1369);
Group action invariants
Degree $n$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $1369$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^4:D_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $16$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,28)(2,27)(3,26)(4,25)(5,14)(6,13)(7,16)(8,15)(9,20)(10,19)(11,18)(12,17)(21,32)(22,31)(23,30)(24,29), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,15)(10,16)(11,13)(12,14)(25,29)(26,30)(27,31)(28,32), (1,16,9,22)(2,15,10,21)(3,14,11,24)(4,13,12,23)(5,26,29,18)(6,25,30,17)(7,28,31,20)(8,27,32,19), (13,19)(14,20)(15,17)(16,18)(21,25)(22,26)(23,27)(24,28), (1,24,5,20)(2,23,6,19)(3,22,7,18)(4,21,8,17)(9,14,29,28)(10,13,30,27)(11,16,31,26)(12,15,32,25) | 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 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 $64$: 16T105 x 12, 32T273 x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 4: $C_2^2$ x 7, $D_{4}$ x 12
Degree 8: $C_2^3$, $D_4\times C_2$ x 18, $C_2^2 \wr C_2$ x 16
Degree 16: $C_2^2 \times D_4$ x 3, 16T105 x 12
Low degree siblings
32T1369 x 127Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 56 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: | 128.2163 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);