Show commands:
Magma
magma: G := TransitiveGroup(16, 1374);
Group action invariants
| Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1374$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $(C_2^2\times D_4^2).D_4$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,11,8,9,5,16,3,13)(2,12,7,10,6,15,4,14), (1,8,6,3,2,7,5,4)(9,14)(10,13)(11,16,12,15), (1,16,8,13,6,11,3,10)(2,15,7,14,5,12,4,9) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_4$ x 4, $C_2^2$ x 7 $8$: $D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$ $16$: $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$ $32$: $Z_8 : Z_8^\times$, $C_4\wr C_2$ x 2, $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$, 16T32 $64$: $((C_8 : C_2):C_2):C_2$ x 2, 16T76, 16T111, 32T264 $128$: 16T227, 16T234, 16T254 $256$: 32T4016 $512$: 16T834 $1024$: 32T64119 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $C_4\wr C_2$
Low degree siblings
16T1374 x 3, 16T1377 x 4, 32T98268 x 4, 32T98269 x 8, 32T98270 x 2, 32T98271 x 4, 32T98272 x 2, 32T98273 x 4, 32T98274 x 4, 32T98275 x 4, 32T98305 x 2, 32T98306 x 4, 32T98307 x 2, 32T98308 x 4, 32T116496 x 2, 32T116499 x 2, 32T116500 x 4, 32T145256 x 2, 32T145259 x 2, 32T145278 x 2, 32T180120 x 2, 32T181304 x 2, 32T192455 x 2, 32T192507 x 2, 32T202622 x 2, 32T202718 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 53 conjugacy class representatives for $(C_2^2\times D_4^2).D_4$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2048=2^{11}$ | 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: | $7$ | ||
| Label: | 2048.cna | magma: IdentifyGroup(G);
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| Character table: | 53 x 53 character table |
magma: CharacterTable(G);