Show commands:
Magma
magma: G := TransitiveGroup(16, 1354);
Group action invariants
| Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1354$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^5.C_2\wr C_4$ | ||
| 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: | (7,8)(9,11,10,12)(15,16), (1,12)(2,11)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,13,11,6,3,16,10,7,2,14,12,5,4,15,9,8) | 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 8, $C_4\times C_2$ x 6, $C_2^3$ $16$: $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$ $32$: $C_2^2 \wr C_2$, $C_2^3 : C_4 $ x 4, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37 $64$: $((C_8 : C_2):C_2):C_2$ x 4, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T76 x 2, 32T239 $128$: 16T208, 16T218, 16T219 x 2, 16T227 x 2, 16T230 $256$: 32T3729, 32T4019 x 2 $512$: 16T912 $1024$: 32T63476 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 8: $((C_8 : C_2):C_2):C_2$
Low degree siblings
16T1354 x 15, 32T98041 x 8, 32T98042 x 16, 32T98043 x 16, 32T98044 x 16, 32T98045 x 8, 32T98046 x 8, 32T98047 x 16, 32T98048 x 8, 32T98049 x 8, 32T98050 x 8, 32T98051 x 8, 32T110365 x 8, 32T141675 x 4, 32T182445 x 4, 32T182576 x 4Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 59 conjugacy class representatives for $C_2^5.C_2\wr C_4$
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: | $6$ | ||
| Label: | 2048.cmi | magma: IdentifyGroup(G);
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| Character table: | 59 x 59 character table |
magma: CharacterTable(G);