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Magma
magma: G := TransitiveGroup(39, 43);
Group action invariants
Degree $n$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $43$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\PSL(3,3)$ | ||
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: | (1,8,18,13,5,20,37,36)(2,9,16,14,4,19,39,34)(3,7,17,15,6,21,38,35)(10,24,33,27)(11,23,32,26,12,22,31,25)(28,30), (1,37,25,7)(2,39,26,9,3,38,27,8)(4,13,22,35,30,19,33,18)(5,14,23,36,28,20,31,16)(6,15,24,34,29,21,32,17)(11,12) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 13: $\PSL(3,3)$
Low degree siblings
13T7 x 2, 26T39 x 2, 39T43Siblings 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 | $1^{39}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{16},1^{7}$ | $117$ | $2$ | $16$ | $( 1, 2)( 4, 5)( 7,19)( 8,20)( 9,21)(11,12)(13,17)(14,18)(15,16)(25,31)(26,32)(27,33)(28,29)(34,37)(35,39)(36,38)$ |
3A | $3^{12},1^{3}$ | $104$ | $3$ | $24$ | $( 4,11,28)( 5,12,29)( 6,10,30)( 7,38,27)( 8,37,26)( 9,39,25)(13,14,15)(16,17,18)(19,36,33)(20,34,32)(21,35,31)(22,24,23)$ |
3B | $3^{13}$ | $624$ | $3$ | $26$ | $( 1,36,31)( 2,34,32)( 3,35,33)( 4, 7,37)( 5, 8,39)( 6, 9,38)(10,17,30)(11,18,28)(12,16,29)(13,25,24)(14,27,23)(15,26,22)(19,20,21)$ |
4A | $4^{8},2^{2},1^{3}$ | $702$ | $4$ | $26$ | $( 1,35,16,26)( 2,36,17,25)( 3,34,18,27)( 4,38,31,12)( 5,39,32,11)( 6,37,33,10)( 7,20)( 8,19, 9,21)(22,29)(23,28,24,30)$ |
6A | $6^{5},3^{2},2,1$ | $936$ | $6$ | $30$ | $( 1, 2)( 4,29,11, 5,28,12)( 6,30,10)( 7,33,38,19,27,36)( 8,32,37,20,26,34)( 9,31,39,21,25,35)(13,16,14,17,15,18)(22,23,24)$ |
8A1 | $8^{4},4,2,1$ | $702$ | $8$ | $32$ | $( 1,11,35, 5,16,39,26,32)( 2,12,36, 4,17,38,25,31)( 3,10,34, 6,18,37,27,33)( 7,22,20,29)( 8,23,19,28, 9,24,21,30)(13,15)$ |
8A-1 | $8^{4},4,2,1$ | $702$ | $8$ | $32$ | $( 1,39,35,32,16,11,26, 5)( 2,38,36,31,17,12,25, 4)( 3,37,34,33,18,10,27, 6)( 7,22,20,29)( 8,24,19,30, 9,23,21,28)(13,15)$ |
13A1 | $13^{3}$ | $432$ | $13$ | $36$ | $( 1,36,25,17,31,14,20,30, 6,39, 7,11,23)( 2,34,27,16,32,13,21,28, 4,38, 9,12,22)( 3,35,26,18,33,15,19,29, 5,37, 8,10,24)$ |
13A-1 | $13^{3}$ | $432$ | $13$ | $36$ | $( 1,30,36, 6,25,39,17, 7,31,11,14,23,20)( 2,28,34, 4,27,38,16, 9,32,12,13,22,21)( 3,29,35, 5,26,37,18, 8,33,10,15,24,19)$ |
13A2 | $13^{3}$ | $432$ | $13$ | $36$ | $( 1,14, 7,25,30,23,31,39,36,20,11,17, 6)( 2,13, 9,27,28,22,32,38,34,21,12,16, 4)( 3,15, 8,26,29,24,33,37,35,19,10,18, 5)$ |
13A-2 | $13^{3}$ | $432$ | $13$ | $36$ | $( 1, 7,30,31,36,11, 6,14,25,23,39,20,17)( 2, 9,28,32,34,12, 4,13,27,22,38,21,16)( 3, 8,29,33,35,10, 5,15,26,24,37,19,18)$ |
Malle's constant $a(G)$: $1/16$
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);