Show commands:
Magma
magma: G := TransitiveGroup(45, 144);
Group action invariants
| Degree $n$: | $45$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $144$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{45}:C_{12}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,22,10,5)(2,24,12,6)(3,23,11,4)(7,26,34,17)(8,27,36,16)(9,25,35,18)(13,28)(14,30)(15,29)(19,31,37,41)(20,32,39,40)(21,33,38,42)(43,44), (1,42,44,20,32,26,29,5,16,10,15,36,3,40,45,21,31,27,30,6,18,11,13,34,2,41,43,19,33,25,28,4,17,12,14,35)(7,38,22,9,37,24,8,39,23) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$, $C_6$ x 3 $8$: $C_4\times C_2$ $12$: $D_{6}$, $C_{12}$ x 2, $C_6\times C_2$ $18$: $S_3\times C_3$ $20$: $F_5$ $24$: $S_3 \times C_4$, 24T2 $36$: $C_6\times S_3$ $40$: $F_{5}\times C_2$ $54$: $(C_9:C_3):C_2$ $60$: $F_5\times C_3$ $72$: 24T65 $108$: 18T45 $120$: $F_5 \times S_3$, 30T26 $216$: 36T214 $360$: 30T91 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 5: $F_5$
Degree 9: $(C_9:C_3):C_2$
Degree 15: $F_5 \times S_3$
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 50 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1080=2^{3} \cdot 3^{3} \cdot 5$ | 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: | 1080.271 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);