Show commands:
Magma
magma: G := TransitiveGroup(13, 7);
Group action invariants
| Degree $n$: | $13$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $7$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $\PSL(3,3)$ | ||
| CHM label: | $L(13)=PSL(3,3)$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | 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: | (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,12)(4,11)(5,6)(7,10) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
13T7, 26T39 x 2, 39T43 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
Conjugacy classes
| Label | Cycle Type | Size | Order | Index | Representative |
| 1A | $1^{13}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{4},1^{5}$ | $117$ | $2$ | $4$ | $( 1, 7)( 4, 6)( 8,13)( 9,10)$ |
| 3A | $3^{3},1^{4}$ | $104$ | $3$ | $6$ | $( 1,13, 9)( 2, 5,11)( 7, 8,10)$ |
| 3B | $3^{4},1$ | $624$ | $3$ | $8$ | $( 1,11, 9)( 2, 6, 5)( 3, 7, 8)( 4,13,12)$ |
| 4A | $4^{2},2^{2},1$ | $702$ | $4$ | $8$ | $( 1, 8, 2,11)( 3, 7, 6,13)( 5,12)( 9,10)$ |
| 6A | $6,3,2,1^{2}$ | $936$ | $6$ | $8$ | $( 1,10,13, 7, 9, 8)( 2,11, 5)( 4, 6)$ |
| 8A1 | $8,4,1$ | $702$ | $8$ | $10$ | $( 1,13, 8, 3, 2, 7,11, 6)( 5, 9,12,10)$ |
| 8A-1 | $8,4,1$ | $702$ | $8$ | $10$ | $( 1, 7, 8, 6, 2,13,11, 3)( 5, 9,12,10)$ |
| 13A1 | $13$ | $432$ | $13$ | $12$ | $( 1, 9, 2,12, 3, 7,13, 6, 8, 4,10, 5,11)$ |
| 13A-1 | $13$ | $432$ | $13$ | $12$ | $( 1, 6, 9, 8, 2, 4,12,10, 3, 5, 7,11,13)$ |
| 13A2 | $13$ | $432$ | $13$ | $12$ | $( 1, 7,10, 2, 6,11, 3, 4, 9,13, 5,12, 8)$ |
| 13A-2 | $13$ | $432$ | $13$ | $12$ | $( 1,10, 6, 3, 9, 5, 8, 7, 2,11, 4,13,12)$ |
Malle's constant $a(G)$: $1/4$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $5616=2^{4} \cdot 3^{3} \cdot 13$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 5616.a | magma: IdentifyGroup(G);
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| Character table: |
| 1A | 2A | 3A | 3B | 4A | 6A | 8A1 | 8A-1 | 13A1 | 13A-1 | 13A2 | 13A-2 | ||
| Size | 1 | 117 | 104 | 624 | 702 | 936 | 702 | 702 | 432 | 432 | 432 | 432 | |
| 2 P | 1A | 1A | 3A | 3B | 2A | 3A | 4A | 4A | 13A1 | 13A-2 | 13A2 | 13A-1 | |
| 3 P | 1A | 2A | 1A | 1A | 4A | 2A | 8A1 | 8A-1 | 13A-2 | 13A-1 | 13A1 | 13A2 | |
| 13 P | 1A | 2A | 3A | 3B | 4A | 6A | 8A-1 | 8A1 | 1A | 1A | 1A | 1A | |
| Type |
magma: CharacterTable(G);