Show commands:
Magma
magma: G := TransitiveGroup(35, 36);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $A_8$ | ||
| 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: | (1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33), (1,3,7)(2,6,5)(9,10,12)(11,14,13)(16,18,21)(17,22,23)(19,26,24)(25,31,28)(27,33,29)(30,35,32) | magma: Generators(G);
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Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 5: None
Degree 7: None
Low degree siblings
8T49, 15T72 x 2, 28T433Siblings 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 | Representative |
| $1^{35}$ | $1$ | $1$ | $()$ | |
| $7^{5}$ | $2880$ | $7$ | $( 1, 4,11,28,15, 2,20)( 3,34,24,31,18,10,13)( 5,25,35,16,14,22, 8) ( 6,23,29,21,33,26, 7)( 9,27,32,30,19,17,12)$ | |
| $7^{5}$ | $2880$ | $7$ | $( 1,20, 2,15,28,11, 4)( 3,13,10,18,31,24,34)( 5, 8,22,14,16,35,25) ( 6, 7,26,33,21,29,23)( 9,12,17,19,30,32,27)$ | |
| $2^{12},1^{11}$ | $105$ | $2$ | $( 1, 6)( 3, 5)( 4,22)( 8,13)(10,30)(11,19)(14,24)(15,26)(16,33)(17,25)(23,31) (28,34)$ | |
| $4^{6},2^{4},1^{3}$ | $1260$ | $4$ | $( 1, 5, 6, 3)( 4,19,22,11)( 8,17,13,25)( 9,18)(10,16,30,33)(12,27) (14,28,24,34)(15,31,26,23)(21,35)(29,32)$ | |
| $2^{14},1^{7}$ | $210$ | $2$ | $( 1, 3)( 2,22)( 4,20)( 5,23)( 6,17)( 9,24)(10,26)(11,21)(12,19)(13,16)(14,18) (25,31)(27,29)(32,35)$ | |
| $4^{7},2^{3},1$ | $2520$ | $4$ | $( 1,26, 3,10)( 2,35,22,32)( 4,31,20,25)( 5,19,23,12)( 6, 9,17,24)( 7, 8) (11,18,21,14)(13,27,16,29)(15,33)(30,34)$ | |
| $3^{10},1^{5}$ | $112$ | $3$ | $( 1,29,31)( 2,16, 4)( 3,27,25)( 5,24,21)( 6,19,18)( 7, 8,33)( 9,11,23) (12,14,17)(13,20,22)(15,34,30)$ | |
| $6^{4},3^{2},2^{2},1$ | $1680$ | $6$ | $( 1,25,29, 3,31,27)( 2,20,16,22, 4,13)( 5,11,24,23,21, 9)( 6,14,19,17,18,12) ( 7,33, 8)(10,26)(15,30,34)(32,35)$ | |
| $3^{11},1^{2}$ | $1120$ | $3$ | $( 1,25,34)( 2, 7,20)( 3,23,22)( 4, 5,31)( 6,17,28)( 8,11,30)(10,13,19) (12,27,35)(14,16,26)(15,24,33)(18,32,29)$ | |
| $6^{4},3^{3},1^{2}$ | $3360$ | $6$ | $( 1,25,34)( 2,28,20,17, 7, 6)( 3, 4,22,31,23, 5)( 8,12,30,35,11,27)(10,13,19) (14,16,26)(15,18,33,29,24,32)$ | |
| $5^{7}$ | $1344$ | $5$ | $( 1,31, 7,17, 5)( 2,28,25,23,34)( 3,20, 6,22, 4)( 8,14,26,12,21) ( 9,15,19,13,18)(10,32,24,29,33)(11,35,30,16,27)$ | |
| $15^{2},5$ | $1344$ | $15$ | $( 1, 7, 5,31,17)( 2,33,27,28,10,11,25,32,35,23,24,30,34,29,16)( 3,26, 9,20,12, 15, 6,21,19,22, 8,13, 4,14,18)$ | |
| $15^{2},5$ | $1344$ | $15$ | $( 1, 7, 5,31,17)( 2,30,32,28,16,24,25,27,29,23,11,33,34,35,10)( 3,13,21,20,18, 8, 6, 9,14,22,15,26, 4,19,12)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$ | 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: | 20160.a | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 7 P | |
| Type |
magma: CharacterTable(G);