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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
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $117$ | $2$ | $( 1,11)( 2, 6)( 3, 8)( 4,13)$ |
$ 4, 4, 2, 2, 1 $ | $702$ | $4$ | $( 1, 2,11, 6)( 3,13, 8, 4)( 5,10)( 9,12)$ |
$ 8, 4, 1 $ | $702$ | $8$ | $( 1, 4, 6, 8,11,13, 2, 3)( 5, 9,10,12)$ |
$ 8, 4, 1 $ | $702$ | $8$ | $( 1, 3, 2,13,11, 8, 6, 4)( 5,12,10, 9)$ |
$ 3, 3, 3, 1, 1, 1, 1 $ | $104$ | $3$ | $( 1, 4,10)( 6, 7, 9)( 8,13,12)$ |
$ 3, 3, 3, 3, 1 $ | $624$ | $3$ | $( 1,13, 7)( 3, 5,11)( 4,12, 9)( 6,10, 8)$ |
$ 6, 3, 2, 1, 1 $ | $936$ | $6$ | $( 1, 3, 9)( 2,10, 8,11,12, 6)( 4, 7)$ |
$ 13 $ | $432$ | $13$ | $( 1, 7, 8, 4, 2, 6, 5,13,12, 3,11,10, 9)$ |
$ 13 $ | $432$ | $13$ | $( 1,12, 4,10, 5, 7, 3, 2, 9,13, 8,11, 6)$ |
$ 13 $ | $432$ | $13$ | $( 1, 9,10,11, 3,12,13, 5, 6, 2, 4, 8, 7)$ |
$ 13 $ | $432$ | $13$ | $( 1, 6,11, 8,13, 9, 2, 3, 7, 5,10, 4,12)$ |
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: |
2 4 4 3 3 3 . . . . 1 1 . 3 3 1 . . . . . . . 3 1 2 13 1 . . . . 1 1 1 1 . . . 1a 2a 4a 8a 8b 13a 13b 13c 13d 3a 6a 3b 2P 1a 1a 2a 4a 4a 13d 13a 13b 13c 3a 3a 3b 3P 1a 2a 4a 8a 8b 13a 13b 13c 13d 1a 2a 1a 5P 1a 2a 4a 8b 8a 13d 13a 13b 13c 3a 6a 3b 7P 1a 2a 4a 8b 8a 13b 13c 13d 13a 3a 6a 3b 11P 1a 2a 4a 8a 8b 13b 13c 13d 13a 3a 6a 3b 13P 1a 2a 4a 8b 8a 1a 1a 1a 1a 3a 6a 3b X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 12 4 . . . -1 -1 -1 -1 3 1 . X.3 13 -3 1 -1 -1 . . . . 4 . 1 X.4 16 . . . . B /C /B C -2 . 1 X.5 16 . . . . /B C B /C -2 . 1 X.6 16 . . . . C B /C /B -2 . 1 X.7 16 . . . . /C /B C B -2 . 1 X.8 26 2 2 . . . . . . -1 -1 -1 X.9 26 -2 . A -A . . . . -1 1 -1 X.10 26 -2 . -A A . . . . -1 1 -1 X.11 27 3 -1 -1 -1 1 1 1 1 . . . X.12 39 -1 -1 1 1 . . . . 3 -1 . A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 B = E(13)^2+E(13)^5+E(13)^6 C = E(13)^4+E(13)^10+E(13)^12 |
magma: CharacterTable(G);