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Magma
magma: G := TransitiveGroup(28, 70);
Group action invariants
Degree $n$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $70$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\PSL(2,8)$ | ||
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: | (1,21,27,15,25,10,24,7,20)(2,19,8,26,17,16,6,5,3)(9,28,18,13,11,22,14,23,12), (1,28,21,4,25,18,10)(2,26,19,17,12,7,11)(3,24,20,16,13,14,9)(5,6,23,15,8,27,22) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: None
Degree 7: None
Degree 14: None
Low degree siblings
9T27, 36T712Siblings 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 | Index | Representative |
1A | $1^{28}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{12},1^{4}$ | $63$ | $2$ | $12$ | $( 1, 2)( 3, 7)( 4, 6)( 8,23)( 9,18)(10,20)(11,19)(12,21)(13,15)(17,24)(25,26)(27,28)$ |
3A | $3^{9},1$ | $56$ | $3$ | $18$ | $( 1,21,17)( 2,26,27)( 3,15,25)( 4, 8,19)( 5,10,12)( 6, 9,18)( 7,11,22)(13,20,16)(14,23,24)$ |
7A1 | $7^{4}$ | $72$ | $7$ | $24$ | $( 1,10,26,23,14,22,19)( 2, 7,17,25,13, 9,21)( 3, 5, 4, 6,16,24,20)( 8,11,15,28,18,27,12)$ |
7A2 | $7^{4}$ | $72$ | $7$ | $24$ | $( 1,26,14,19,10,23,22)( 2,17,13,21, 7,25, 9)( 3, 4,16,20, 5, 6,24)( 8,15,18,12,11,28,27)$ |
7A3 | $7^{4}$ | $72$ | $7$ | $24$ | $( 1,23,19,26,22,10,14)( 2,25,21,17, 9, 7,13)( 3, 6,20, 4,24, 5,16)( 8,28,12,15,27,11,18)$ |
9A1 | $9^{3},1$ | $56$ | $9$ | $24$ | $( 1,18, 3,17, 9,25,21, 6,15)( 2, 7,14,27,22,24,26,11,23)( 4,10,13,19, 5,16, 8,12,20)$ |
9A2 | $9^{3},1$ | $56$ | $9$ | $24$ | $( 1, 3, 9,21,15,18,17,25, 6)( 2,14,22,26,23, 7,27,24,11)( 4,13, 5, 8,20,10,19,16,12)$ |
9A4 | $9^{3},1$ | $56$ | $9$ | $24$ | $( 1,25,18,21, 3, 6,17,15, 9)( 2,24, 7,26,14,11,27,23,22)( 4,16,10, 8,13,12,19,20, 5)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Group invariants
Order: | $504=2^{3} \cdot 3^{2} \cdot 7$ | 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: | 504.156 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A | 7A1 | 7A2 | 7A3 | 9A1 | 9A2 | 9A4 | ||
Size | 1 | 63 | 56 | 72 | 72 | 72 | 56 | 56 | 56 | |
2 P | 1A | 1A | 3A | 7A2 | 7A3 | 7A1 | 9A1 | 9A2 | 9A4 | |
3 P | 1A | 2A | 1A | 7A3 | 7A1 | 7A2 | 3A | 3A | 3A | |
7 P | 1A | 2A | 3A | 1A | 1A | 1A | 9A1 | 9A2 | 9A4 | |
Type | ||||||||||
504.156.1a | R | |||||||||
504.156.7a | R | |||||||||
504.156.7b1 | R | |||||||||
504.156.7b2 | R | |||||||||
504.156.7b3 | R | |||||||||
504.156.8a | R | |||||||||
504.156.9a1 | R | |||||||||
504.156.9a2 | R | |||||||||
504.156.9a3 | R |
magma: CharacterTable(G);