Show commands:
Magma
magma: G := TransitiveGroup(36, 85036);
Group action invariants
| Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $85036$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^8.((C_3:S_3)^3.C_2^3)$ | ||
| 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: | (1,22,14,10,27,35)(2,23,15,12,25,36,3,24,13,11,26,34)(4,9,28,19,16,32)(5,8,29,20,18,31,6,7,30,21,17,33), (1,13,26,3,14,25,2,15,27)(4,6,5)(7,21)(8,19)(9,20)(16,17,18)(22,34,24,36,23,35)(31,33,32), (1,28,2,29,3,30)(4,25,17,15,5,27,16,13,6,26,18,14)(7,24,31,35)(8,22,32,36)(9,23,33,34)(10,19,12,21,11,20), (1,7,13,20)(2,9,14,21)(3,8,15,19)(4,34,18,10,5,35,16,12,6,36,17,11)(22,30)(23,28,24,29)(25,31)(26,33,27,32) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 15 $4$: $C_4$ x 8, $C_2^2$ x 35 $8$: $C_4\times C_2$ x 28, $C_2^3$ x 15 $16$: $C_4\times C_2^2$ x 14, $C_2^4$ $32$: $Q_8:C_2^2$ x 2, 32T34 $36$: $C_3^2:C_4$ $64$: 16T68 $72$: 12T40 x 7 $144$: 24T241 x 7 $576$: 24T1394 $2592$: 12T242 $5184$: 24T7644 $46656$: 24T14743 $3779136$: 36T50018 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: None
Degree 4: $C_2^2$
Degree 6: None
Degree 9: None
Degree 12: 12T242
Degree 18: None
Low degree siblings
36T85036 x 8Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 2,062 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $306110016=2^{6} \cdot 3^{14}$ | 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: | 306110016.fu | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);