Show commands:
Magma
magma: G := TransitiveGroup(32, 12882);
Group action invariants
| Degree $n$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $12882$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_4^2:C_2^3$ | ||
| 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: | (1,28,17,10)(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);
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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 $64$: $(((C_4 \times C_2): C_2):C_2):C_2$ 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 384Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 80 conjugacy class representatives for $D_4^2:C_2^3$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $512=2^{9}$ | 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: | $4$ | ||
| Label: | 512.7530050 | magma: IdentifyGroup(G);
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| Character table: | 80 x 80 character table |
magma: CharacterTable(G);