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Magma
magma: G := TransitiveGroup(35, 44);
Group invariants
| Abstract group: | $S_8$ | magma: IdentifyGroup(G);
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| Order: | $40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$ | 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 | magma: NilpotencyClass(G);
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Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $44$ | 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: | $(1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33)$, $(1,3)(5,6)(9,12)(11,14)(17,23)(18,21)(19,24)(25,31)(27,29)(32,35)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: None
Degree 7: None
Low degree siblings
8T50, 16T1838, 28T502, 30T1153Siblings 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 | $1^{35}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{10},1^{15}$ | $28$ | $2$ | $10$ | $( 1,28)( 4, 5)( 7,25)( 8,27)(10,11)(12,13)(15,32)(16,24)(17,20)(18,26)$ |
| 2B | $2^{12},1^{11}$ | $105$ | $2$ | $12$ | $( 1,16)( 3,30)( 4,11)( 5,10)( 6,33)( 8,13)(12,27)(14,34)(19,22)(21,35)(23,31)(24,28)$ |
| 2C | $2^{14},1^{7}$ | $210$ | $2$ | $14$ | $( 1,28)( 3,22)( 4, 5)( 6,34)( 7,20)( 8,13)(10,11)(12,27)(14,33)(15,26)(16,24)(17,25)(18,32)(19,30)$ |
| 2D | $2^{16},1^{3}$ | $420$ | $2$ | $16$ | $( 1,16)( 3,30)( 4, 5)( 6,34)( 7,15)( 8,13)(10,11)(12,27)(14,33)(17,18)(19,22)(20,26)(21,31)(23,35)(24,28)(25,32)$ |
| 3A | $3^{10},1^{5}$ | $112$ | $3$ | $20$ | $( 1,28,22)( 4,34, 5)( 7,25,23)( 8,27, 9)(10,33,11)(12,29,13)(15,32,21)(16,30,24)(17,31,20)(18,35,26)$ |
| 3B | $3^{11},1^{2}$ | $1120$ | $3$ | $22$ | $( 1,27,30)( 2, 3,19)( 4,23,26)( 5,25,35)( 7,18,34)( 8,16,22)( 9,24,28)(10,21,20)(11,32,31)(12,29,13)(15,17,33)$ |
| 4A | $4^{7},2^{3},1$ | $420$ | $4$ | $24$ | $( 1,28,22, 3)( 4,34, 6, 5)( 7,31,20,23)( 8,29,13, 9)(10,33,14,11)(12,27)(15,35,26,21)(16,30,19,24)(17,25)(18,32)$ |
| 4B | $4^{6},2^{4},1^{3}$ | $1260$ | $4$ | $22$ | $( 1,16,33, 6)( 3,24,10,34)( 4,22,19,11)( 5,28,30,14)( 7,29)( 8,23)( 9,20)(12,17,27,25)(13,31)(15,21,26,35)$ |
| 4C | $4^{7},2,1^{5}$ | $1260$ | $4$ | $22$ | $( 1,16,22,19)( 3,24,28,30)( 4,33, 6,11)( 5,10,34,14)( 7,23,20,31)( 8,29,13, 9)(12,27)(15,21,26,35)$ |
| 4D | $4^{7},2^{3},1$ | $2520$ | $4$ | $24$ | $( 1,16,22,19)( 3,24,28,30)( 4,34, 6, 5)( 7,21,20,35)( 8,29,13, 9)(10,33,14,11)(12,27)(15,23,26,31)(17,18)(25,32)$ |
| 5A | $5^{7}$ | $1344$ | $5$ | $28$ | $( 1,28,22, 3, 2)( 4,17,25,34, 7)( 5,31,20,23, 6)( 8,16,12,27,30)( 9,19,24,29,13)(10,18,32,33,15)(11,35,26,21,14)$ |
| 6A | $6^{3},3^{4},2,1^{3}$ | $1120$ | $6$ | $24$ | $( 1,28,22)( 2, 3)( 4,23, 5,25,34, 7)( 8,16, 9,24,27,30)(10,21,11,32,33,15)(12,29,13)(17,31,20)(18,35,26)$ |
| 6B | $6^{3},3^{5},2$ | $1120$ | $6$ | $26$ | $( 1,27,30)( 2, 3,19)( 4,21,26,10,23,20)( 5,32,35,11,25,31)( 6,14)( 7,17,34,15,18,33)( 8,16,22)( 9,24,28)(12,29,13)$ |
| 6C | $6^{4},3^{2},2^{2},1$ | $1680$ | $6$ | $26$ | $( 1,28,22)( 2, 3)( 4,23, 5,25,34, 7)( 8,10, 9,11,27,33)(12,35,13,18,29,26)(14,19)(15,16,21,24,32,30)(17,31,20)$ |
| 6D | $6^{4},3^{3},1^{2}$ | $3360$ | $6$ | $26$ | $( 1,29,13, 9,19, 2)( 3,24,28,12,27,30)( 4,18,33, 7,23,20)( 5,35,14)( 6,11,31)( 8,16,22)(10,17,34,15,21,26)$ |
| 6E | $6^{5},3,2$ | $3360$ | $6$ | $28$ | $( 1,29,13, 9,19, 2)( 3,24,28,12,27,30)( 4,17,33,15,23,26)( 5,31,14,11,35, 6)( 7,21,20,10,18,34)( 8,16,22)(25,32)$ |
| 7A | $7^{5}$ | $5760$ | $7$ | $30$ | $( 1,29, 6, 5,31,14, 2)( 3,24,35,13, 4,17,33)( 7,21,19,11,28,12,34)( 8,23,26, 9,20,10,22)(15,16,18,27,25,32,30)$ |
| 8A | $8^{3},4^{2},2,1$ | $5040$ | $8$ | $28$ | $( 1,29, 6, 2)( 3,24,35,14, 5,28,12,34)( 4,22, 8,23,19,11,31,13)( 7,16,18,33)( 9,20)(10,17,27,25,30,15,21,26)$ |
| 10A | $10^{2},5^{3}$ | $4032$ | $10$ | $30$ | $( 1,28,22, 3, 2)( 4,17,25,34, 7)( 5,31,20,23, 6)( 8,10,12,32,30,15,16,18,27,33)( 9,14,24,35,13,21,19,11,29,26)$ |
| 12A | $12^{2},6,4,1$ | $3360$ | $12$ | $30$ | $( 1,16,33, 3,24,10,22,19,11,28,30,14)( 4,34, 6, 5)( 7,21, 8,23,26, 9,20,35,13,31,15,29)(12,17,18,27,25,32)$ |
| 15A | $15^{2},5$ | $2688$ | $15$ | $32$ | $( 1,28,22, 3, 2)( 4,12,32,34, 8,10,17,27,33, 7,16,18,25,30,15)( 5,29,26,23,19,11,31,13,21, 6,24,35,20, 9,14)$ |
Malle's constant $a(G)$: $1/10$
magma: ConjugacyClasses(G);
Character table
| 1A | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 6A | 6B | 6C | 6D | 6E | 7A | 8A | 10A | 12A | 15A | ||
| Size | 1 | 28 | 105 | 210 | 420 | 112 | 1120 | 420 | 1260 | 1260 | 2520 | 1344 | 1120 | 1120 | 1680 | 3360 | 3360 | 5760 | 5040 | 4032 | 3360 | 2688 | |
| 2 P | 1A | 1A | 1A | 1A | 1A | 3A | 3B | 2C | 2B | 2C | 2C | 5A | 3A | 3B | 3A | 3B | 3B | 7A | 4B | 5A | 6C | 15A | |
| 3 P | 1A | 2A | 2B | 2C | 2D | 1A | 1A | 4A | 4B | 4C | 4D | 5A | 2A | 2A | 2C | 2B | 2D | 7A | 8A | 10A | 4A | 5A | |
| 5 P | 1A | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 1A | 6A | 6B | 6C | 6D | 6E | 7A | 8A | 2A | 12A | 3A | |
| 7 P | 1A | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 6A | 6B | 6C | 6D | 6E | 1A | 8A | 10A | 12A | 15A | |
| Type |
magma: CharacterTable(G);
Regular extensions
Data not computed