Show commands:
Magma
magma: G := TransitiveGroup(40, 214359);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $214359$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2^{10}.(C_4^4.C_2^6:F_5)$ | ||
| 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: | (1,34,17,28,4,35,19,26,2,33,18,27,3,36,20,25)(5,38,23,31,8,39,21,29,6,37,24,32,7,40,22,30)(9,14,12,15)(10,13,11,16), (3,4)(5,7,6,8)(9,24,38,29,15,18,34,27,10,23,37,30,16,17,33,28)(11,22,39,32,14,20,36,25)(12,21,40,31,13,19,35,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $D_{4}$ x 2, $C_4\times C_2$ $16$: $C_2^2:C_4$ $20$: $F_5$ $40$: $F_{5}\times C_2$ $320$: $(C_2^4 : C_5):C_4$ x 3 $640$: $((C_2^4 : C_5):C_4)\times C_2$ x 3 Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: None
Degree 4: None
Degree 5: $F_5$
Degree 8: None
Degree 10: $(C_2^4 : C_5):C_4$
Degree 20: 20T449
Low degree siblings
There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.
Conjugacy classes
The 2291 conjugacy class representatives for $C_2^{10}.(C_4^4.C_2^6:F_5)$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $335544320=2^{26} \cdot 5$ | 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: | 335544320.bgj | magma: IdentifyGroup(G);
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| Character table: | not computed |
magma: CharacterTable(G);