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Magma
magma: G := TransitiveGroup(36, 49);
Group action invariants
Degree $n$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $49$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $F_9$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,3,2,4)(5,13,18,29,34,25,22,10)(6,14,17,30,33,26,21,9)(7,16,20,31,36,28,24,12)(8,15,19,32,35,27,23,11), (1,22,27,6)(2,21,28,5)(3,24,25,8)(4,23,26,7)(9,35,19,29)(10,36,20,30)(11,34,18,31)(12,33,17,32)(13,14)(15,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $8$: $C_8$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $C_4$
Degree 6: None
Degree 9: $C_3^2:C_8$
Degree 12: 12T46
Degree 18: 18T28
Low degree siblings
9T15, 12T46, 18T28, 24T81Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5,34)( 6,33)( 7,36)( 8,35)( 9,30)(10,29)(11,32)(12,31)(13,25)(14,26)(15,27) (16,28)(17,21)(18,22)(19,23)(20,24)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4, 2, 2 $ | $9$ | $4$ | $( 1, 2)( 3, 4)( 5,18,34,22)( 6,17,33,21)( 7,20,36,24)( 8,19,35,23) ( 9,14,30,26)(10,13,29,25)(11,15,32,27)(12,16,31,28)$ | |
$ 4, 4, 4, 4, 4, 4, 4, 4, 2, 2 $ | $9$ | $4$ | $( 1, 2)( 3, 4)( 5,22,34,18)( 6,21,33,17)( 7,24,36,20)( 8,23,35,19) ( 9,26,30,14)(10,25,29,13)(11,27,32,15)(12,28,31,16)$ | |
$ 8, 8, 8, 8, 4 $ | $9$ | $8$ | $( 1, 3, 2, 4)( 5,13,18,29,34,25,22,10)( 6,14,17,30,33,26,21, 9) ( 7,16,20,31,36,28,24,12)( 8,15,19,32,35,27,23,11)$ | |
$ 8, 8, 8, 8, 4 $ | $9$ | $8$ | $( 1, 3, 2, 4)( 5,25,18,10,34,13,22,29)( 6,26,17, 9,33,14,21,30) ( 7,28,20,12,36,16,24,31)( 8,27,19,11,35,15,23,32)$ | |
$ 8, 8, 8, 8, 4 $ | $9$ | $8$ | $( 1, 4, 2, 3)( 5,10,22,25,34,29,18,13)( 6, 9,21,26,33,30,17,14) ( 7,12,24,28,36,31,20,16)( 8,11,23,27,35,32,19,15)$ | |
$ 8, 8, 8, 8, 4 $ | $9$ | $8$ | $( 1, 4, 2, 3)( 5,29,22,13,34,10,18,25)( 6,30,21,14,33, 9,17,26) ( 7,31,24,16,36,12,20,28)( 8,32,23,15,35,11,19,27)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 5,34)( 2, 6,33)( 3, 7,36)( 4, 8,35)( 9,13,19)(10,14,20)(11,16,18) (12,15,17)(21,27,31)(22,28,32)(23,25,30)(24,26,29)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $72=2^{3} \cdot 3^{2}$ | 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: | 72.39 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A | 4A1 | 4A-1 | 8A1 | 8A-1 | 8A3 | 8A-3 | ||
Size | 1 | 9 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | |
2 P | 1A | 1A | 3A | 2A | 2A | 4A-1 | 4A1 | 4A1 | 4A-1 | |
3 P | 1A | 2A | 1A | 4A-1 | 4A1 | 8A1 | 8A-1 | 8A3 | 8A-3 | |
Type | ||||||||||
72.39.1a | R | |||||||||
72.39.1b | R | |||||||||
72.39.1c1 | C | |||||||||
72.39.1c2 | C | |||||||||
72.39.1d1 | C | |||||||||
72.39.1d2 | C | |||||||||
72.39.1d3 | C | |||||||||
72.39.1d4 | C | |||||||||
72.39.8a | R |
magma: CharacterTable(G);