Show commands:
Magma
magma: G := TransitiveGroup(24, 4994);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $4994$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $(C_2\times D_6^2).D_4$ | ||
| 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: | (1,18,8,22,6,13,12,23,4,15,10,20,2,17,7,21,5,14,11,24,3,16,9,19), (1,18,11,23)(2,17,12,24)(3,13,7,20)(4,14,8,19)(5,15,10,21)(6,16,9,22), (1,9)(2,10)(3,8)(4,7)(5,12)(6,11)(13,17,16)(14,18,15)(19,24,21,20,23,22) | 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 12, $C_4\times C_2$ x 6, $C_2^3$ $16$: $D_4\times C_2$ x 6, $C_2^2:C_4$ x 12, $C_4\times C_2^2$ $32$: $C_2^2 \wr C_2$ x 4, $C_2 \times (C_2^2:C_4)$ x 3 $64$: $(((C_4 \times C_2): C_2):C_2):C_2$, 16T79, 16T146 $72$: $C_3^2:D_4$ $128$: $C_2 \wr C_2\wr C_2$ x 2, 32T1151 $144$: 12T77, 12T79 x 2 $256$: 16T542 $288$: 12T125 x 2, 24T645 $576$: 24T1477 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $D_{4}$
Degree 6: $C_3^2:D_4$
Degree 8: $C_2 \wr C_2\wr C_2$
Degree 12: 12T80
Low degree siblings
24T4994 x 15, 24T5059 x 16Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 78 conjugacy class representatives for $(C_2\times D_6^2).D_4$
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.fp | magma: IdentifyGroup(G);
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| Character table: | 78 x 78 character table |
magma: CharacterTable(G);