Show commands:
Magma
magma: G := TransitiveGroup(24, 4998);
Group action invariants
Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $4998$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $(C_2^2\times C_6^2):\SD_{16}$ | ||
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: | (1,17,2,18)(3,13,6,16)(4,14,5,15)(7,21)(8,22)(9,19,12,24)(10,20,11,23), (1,15,7,22,4,14,11,24)(2,16,8,21,3,13,12,23)(5,17,9,20,6,18,10,19), (1,10)(2,9)(3,8,5,11)(4,7,6,12)(13,23,14,24)(15,22,18,19)(16,21,17,20) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $D_{4}$ x 6, $C_2^3$ $16$: $QD_{16}$ x 2, $D_4\times C_2$ x 3 $32$: $Z_8 : Z_8^\times$, $C_2^2 \wr C_2$, 16T48 $64$: $(((C_4 \times C_2): C_2):C_2):C_2$, 16T138, 16T155 $128$: $C_2 \wr C_2\wr C_2$ x 2, 32T1557 $144$: $(C_3^2:C_8):C_2$ $256$: 16T700 $288$: 18T110 $576$: 24T1460 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $D_{4}$
Degree 6: None
Degree 8: $C_2 \wr C_2\wr C_2$
Degree 12: 12T84
Low degree siblings
24T4998 x 15Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 51 conjugacy class representatives for $(C_2^2\times C_6^2):\SD_{16}$
magma: ConjugacyClasses(G);
Group invariants
Order: | $2304=2^{8} \cdot 3^{2}$ | 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: | not nilpotent | ||
Label: | 2304.fs | magma: IdentifyGroup(G);
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Character table: | 51 x 51 character table |
magma: CharacterTable(G);