Show commands:
Magma
magma: G := TransitiveGroup(40, 218046);
Group action invariants
| Degree $n$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $218046$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_5^4.D_5^4.C_4^2:C_2^2$ | ||
| 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: | (1,31,3,34,4,33,2,35)(5,32)(6,28)(7,29,8,30,10,27,9,26)(11,25,15,22,12,23,13,21)(14,24)(16,37)(17,36,18,40,20,38,19,39), (1,27,3,30,5,28,2,26,4,29)(6,22,7,23,10,21,9,25)(8,24)(11,17,15,16)(12,18,14,20)(13,19)(31,40,35,36,32,39,33,38)(34,37), (1,35,3,31)(2,33)(4,34,5,32)(6,27,8,26)(7,29)(9,28,10,30)(11,21,13,22,15,23,12,24,14,25)(16,40,18,39,20,38,17,37,19,36) | 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 8, $C_2^3$ $16$: $D_4\times C_2$ x 4, $Q_8:C_2$ x 3 $32$: $Z_8 : Z_8^\times$, $C_2^2 \wr C_2$ $64$: $(C_4^2 : C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$ Resolvents shown for degrees $\leq 10$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 5: None
Degree 8: $D_4$
Degree 10: None
Degree 20: None
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 658 conjugacy class representatives for $C_5^4.D_5^4.C_4^2:C_2^2$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $400000000=2^{10} \cdot 5^{8}$ | 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: | 400000000.drv | magma: IdentifyGroup(G);
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| Character table: | not computed |
magma: CharacterTable(G);