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Magma
magma: G := TransitiveGroup(26, 42);
Group invariants
Abstract group: | $\PSL(2,25)$ | magma: IdentifyGroup(G);
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Order: | $7800=2^{3} \cdot 3 \cdot 5^{2} \cdot 13$ | 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 | magma: NilpotencyClass(G);
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Group action invariants
Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $42$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Parity: | $1$ | magma: IsEven(G);
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Primitive: | yes | magma: IsPrimitive(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,19,26,4,17,2,6,22,18,5,21,12,16)(3,8,7,14,23,13,24,10,25,9,11,15,20)$, $(1,21,5,9,4,2,10,16,12,18,14,25,3)(6,23,7,24,17,11,8,20,26,15,22,19,13)$ | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 13: None
Low degree siblings
There are no siblings 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 | $1^{26}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{12},1^{2}$ | $325$ | $2$ | $12$ | $( 1,11)( 3, 7)( 4,16)( 5,15)( 6,18)( 8,26)( 9,13)(10,23)(12,25)(17,19)(20,21)(22,24)$ |
3A | $3^{8},1^{2}$ | $650$ | $3$ | $16$ | $( 2, 9,24)( 3, 5,14)( 4,25,16)( 6,20,17)( 7,10,22)( 8,12,26)(11,18,21)(13,23,15)$ |
4A | $4^{6},1^{2}$ | $650$ | $4$ | $18$ | $( 1,19,11,17)( 3,13, 7, 9)( 4,18,16, 6)( 5,22,15,24)( 8,20,26,21)(10,25,23,12)$ |
5A | $5^{5},1$ | $312$ | $5$ | $20$ | $( 2,15,10, 9,21)( 3,14,16,24,12)( 4,20, 8,11,18)( 5, 6,23,22,17)( 7,19,13,25,26)$ |
5B | $5^{5},1$ | $312$ | $5$ | $20$ | $( 1,12, 8,13, 6)( 3,23,16,21,17)( 4,10, 7,19,20)( 5,14,15,22,24)( 9,26,25,11,18)$ |
6A | $6^{4},1^{2}$ | $650$ | $6$ | $20$ | $( 2,20, 9,17,24, 6)( 3,11, 5,18,14,21)( 4, 8,25,12,16,26)( 7,15,10,13,22,23)$ |
12A1 | $12^{2},1^{2}$ | $650$ | $12$ | $22$ | $( 2,18,20,14, 9,21,17, 3,24,11, 6, 5)( 4, 7, 8,15,25,10,12,13,16,22,26,23)$ |
12A5 | $12^{2},1^{2}$ | $650$ | $12$ | $22$ | $( 2,21, 6,14,24,18,17, 5, 9,11,20, 3)( 4,10,26,15,16, 7,12,23,25,22, 8,13)$ |
13A1 | $13^{2}$ | $600$ | $13$ | $24$ | $( 1, 3, 4,25,11,16,26, 9,14, 5,21,22,24)( 2,20, 7,17, 6,15,19, 8,18,10,23,13,12)$ |
13A2 | $13^{2}$ | $600$ | $13$ | $24$ | $( 1,22, 5, 9,16,25, 3,24,21,14,26,11, 4)( 2,13,10, 8,15,17,20,12,23,18,19, 6, 7)$ |
13A3 | $13^{2}$ | $600$ | $13$ | $24$ | $( 1,21, 9,11, 3,22,14,16, 4,24, 5,26,25)( 2,23, 8, 6,20,13,18,15, 7,12,10,19,17)$ |
13A4 | $13^{2}$ | $600$ | $13$ | $24$ | $( 1,11,14,24,25, 9,22, 4,26,21, 3,16, 5)( 2, 6,18,12,17, 8,13, 7,19,23,20,15,10)$ |
13A5 | $13^{2}$ | $600$ | $13$ | $24$ | $( 1,14,25,22,26, 3, 5,11,24, 9, 4,21,16)( 2,18,17,13,19,20,10, 6,12, 8, 7,23,15)$ |
13A6 | $13^{2}$ | $600$ | $13$ | $24$ | $( 1, 9, 3,14, 4, 5,25,21,11,22,16,24,26)( 2, 8,20,18, 7,10,17,23, 6,13,15,12,19)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 3A | 4A | 5A | 5B | 6A | 12A1 | 12A5 | 13A1 | 13A2 | 13A3 | 13A4 | 13A5 | 13A6 | ||
Size | 1 | 325 | 650 | 650 | 312 | 312 | 650 | 650 | 650 | 600 | 600 | 600 | 600 | 600 | 600 | |
2 P | 1A | 1A | 3A | 2A | 5A | 5B | 3A | 6A | 6A | 13A6 | 13A1 | 13A5 | 13A2 | 13A4 | 13A3 | |
3 P | 1A | 2A | 1A | 4A | 5A | 5B | 2A | 4A | 4A | 13A4 | 13A5 | 13A1 | 13A3 | 13A6 | 13A2 | |
5 P | 1A | 2A | 3A | 4A | 1A | 1A | 6A | 12A5 | 12A1 | 13A2 | 13A4 | 13A6 | 13A5 | 13A3 | 13A1 | |
13 P | 1A | 2A | 3A | 4A | 5A | 5B | 6A | 12A1 | 12A5 | 1A | 1A | 1A | 1A | 1A | 1A | |
Type |
magma: CharacterTable(G);
Regular extensions
Data not computed