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Magma
magma: G := TransitiveGroup(28, 120);
Group invariants
Abstract group: | $\PSL(2,13)$ | magma: IdentifyGroup(G);
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Order: | $1092=2^{2} \cdot 3 \cdot 7 \cdot 13$ | 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 $n$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $120$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | $(1,26,11,15,17,13,10)(2,25,12,16,18,14,9)(3,7,5,28,20,22,23)(4,8,6,27,19,21,24)$, $(1,14,9,17,28,3,20)(2,13,10,18,27,4,19)(5,7,26,15,23,22,11)(6,8,25,16,24,21,12)$ | 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: $\PSL(2,13)$
Low degree siblings
14T30, 42T176Siblings 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 | $2^{14}$ | $91$ | $2$ | $14$ | $( 1, 2)( 3,10)( 4, 9)( 5,25)( 6,26)( 7, 8)(11,16)(12,15)(13,20)(14,19)(17,27)(18,28)(21,23)(22,24)$ |
3A | $3^{8},1^{4}$ | $182$ | $3$ | $16$ | $( 3,21,19)( 4,22,20)( 5,27,12)( 6,28,11)( 9,24,13)(10,23,14)(15,25,17)(16,26,18)$ |
6A | $6^{4},2^{2}$ | $182$ | $6$ | $22$ | $( 1, 2)( 3,14,21,10,19,23)( 4,13,22, 9,20,24)( 5,15,27,25,12,17)( 6,16,28,26,11,18)( 7, 8)$ |
7A1 | $7^{4}$ | $156$ | $7$ | $24$ | $( 1,27,25, 7,14,24,10)( 2,28,26, 8,13,23, 9)( 3,16, 5,21,17,11,19)( 4,15, 6,22,18,12,20)$ |
7A2 | $7^{4}$ | $156$ | $7$ | $24$ | $( 1,25,14,10,27, 7,24)( 2,26,13, 9,28, 8,23)( 3, 5,17,19,16,21,11)( 4, 6,18,20,15,22,12)$ |
7A3 | $7^{4}$ | $156$ | $7$ | $24$ | $( 1, 7,10,25,24,27,14)( 2, 8, 9,26,23,28,13)( 3,21,19, 5,11,16,17)( 4,22,20, 6,12,15,18)$ |
13A1 | $13^{2},1^{2}$ | $84$ | $13$ | $24$ | $( 1,19,23, 6,13,18,12,26, 4,28,21,10, 8)( 2,20,24, 5,14,17,11,25, 3,27,22, 9, 7)$ |
13A2 | $13^{2},1^{2}$ | $84$ | $13$ | $24$ | $( 1,23,13,12, 4,21, 8,19, 6,18,26,28,10)( 2,24,14,11, 3,22, 7,20, 5,17,25,27, 9)$ |
Malle's constant $a(G)$: $1/14$
magma: ConjugacyClasses(G);
Character table
1A | 2A | 3A | 6A | 7A1 | 7A2 | 7A3 | 13A1 | 13A2 | ||
Size | 1 | 91 | 182 | 182 | 156 | 156 | 156 | 84 | 84 | |
2 P | 1A | 1A | 3A | 3A | 7A2 | 7A3 | 7A1 | 13A2 | 13A1 | |
3 P | 1A | 2A | 1A | 2A | 7A3 | 7A1 | 7A2 | 13A1 | 13A2 | |
7 P | 1A | 2A | 3A | 6A | 1A | 1A | 1A | 13A2 | 13A1 | |
13 P | 1A | 2A | 3A | 6A | 7A1 | 7A2 | 7A3 | 1A | 1A | |
Type | ||||||||||
1092.25.1a | R | |||||||||
1092.25.7a1 | R | |||||||||
1092.25.7a2 | R | |||||||||
1092.25.12a1 | R | |||||||||
1092.25.12a2 | R | |||||||||
1092.25.12a3 | R | |||||||||
1092.25.13a | R | |||||||||
1092.25.14a | R | |||||||||
1092.25.14b | R |
magma: CharacterTable(G);
Regular extensions
Data not computed