Show commands:
Magma
magma: G := TransitiveGroup(24, 5029);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $5029$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $A_4^2:D_8$ | ||
| 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: | (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,24,20)(14,23,19)(15,22,18,16,21,17), (1,22,10,19,5,16,3,24,12,17,8,13,2,21,9,20,6,15,4,23,11,18,7,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $8$: $D_{4}$ $12$: $D_{6}$, $C_6\times C_2$ $16$: $D_{8}$ $18$: $S_3\times C_3$ $24$: $(C_6\times C_2):C_2$, $D_4 \times C_3$ $36$: $C_6\times S_3$ $48$: 24T37, 24T40 $72$: 12T42 $144$: 24T246 $288$: $A_4\wr C_2$ $576$: 12T158 $1152$: 12T208 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $D_{4}$
Degree 6: $S_3\times C_3$
Degree 8: None
Degree 12: 12T42
Low degree siblings
24T5029, 32T205447 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 58 conjugacy class representatives for $A_4^2:D_8$
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.gq | magma: IdentifyGroup(G);
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| Character table: | 58 x 58 character table |
magma: CharacterTable(G);