Show commands:
Magma
magma: G := TransitiveGroup(16, 1728);
Group action invariants
| Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1728$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^2.C_2^2\wr D_4$ | ||
| 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: | (1,7,2,8)(3,6,4,5)(9,10)(11,12), (1,10,3,11,2,9,4,12)(5,13,7,15,6,14,8,16), (9,10)(13,14), (1,5,2,6)(3,4), (1,6)(2,5)(3,7)(4,8)(9,10)(11,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 15 $4$: $C_2^2$ x 35 $8$: $D_{4}$ x 28, $C_2^3$ x 15 $16$: $D_4\times C_2$ x 42, $C_2^4$ $32$: $C_2^2 \wr C_2$ x 28, $C_2^2 \times D_4$ x 7 $64$: $(((C_4 \times C_2): C_2):C_2):C_2$ x 6, 16T105 x 7 $128$: $C_2 \wr C_2\wr C_2$ x 12, 16T241 x 3, 16T245 x 3, 16T325 $256$: 16T509 x 6, 32T4223 x 3 $512$: 16T819 x 3, 16T907, 16T919 x 3 $1024$: 32T40151 x 3 $2048$: 16T1340 $4096$: 32T317640 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 8: $C_2 \wr C_2\wr C_2$
Low degree siblings
16T1719 x 16, 16T1728 x 47, 32T399975 x 48, 32T399976 x 24, 32T399977 x 48, 32T399978 x 48, 32T399979 x 24, 32T399980 x 24, 32T399981 x 24, 32T399982 x 24, 32T399983 x 24, 32T399984 x 24, 32T399985 x 24, 32T399986 x 24, 32T399987 x 24, 32T399988 x 48, 32T399989 x 48, 32T399990 x 24, 32T399991 x 8, 32T399992 x 8, 32T399993 x 8, 32T400215 x 24, 32T400216 x 48, 32T400217 x 24, 32T400218 x 24, 32T400219 x 24, 32T400220 x 48, 32T400221 x 48, 32T400222 x 24, 32T400223 x 24, 32T400224 x 24, 32T400225 x 48, 32T400226 x 24, 32T400227 x 48, 32T400228 x 24, 32T400229 x 24, 32T400230 x 24, 32T400231 x 48, 32T400232 x 24, 32T400233 x 48, 32T400234 x 24, 32T400235 x 24, 32T400236 x 48, 32T405528 x 8, 32T431383 x 12, 32T431523 x 12, 32T431732 x 24, 32T431871 x 24, 32T519524 x 24, 32T522319 x 12, 32T549500 x 12, 32T549570 x 12, 32T640154 x 24, 32T715889 x 12Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 152 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $8192=2^{13}$ | 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: | 8192.wd | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);