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Magma
magma: G := TransitiveGroup(35, 34);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_5\times A_7$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $5$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,12,23,19,5,11,22,18,4,15,21,17,3,14,25,16,2,13,24,20)(6,27,8,29,10,26,7,28,9,30)(31,32,33,34,35), (1,35,9,3,32,6,5,34,8,2,31,10,4,33,7)(11,25,19,13,22,16,15,24,18,12,21,20,14,23,17)(26,30,29,28,27) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $5$: $C_5$ $2520$: $A_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $C_5$
Degree 7: $A_7$
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 | Representative |
| $1^{35}$ | $1$ | $1$ | $()$ | |
| $5^{7}$ | $1$ | $5$ | $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$ | |
| $5^{7}$ | $1$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$ | |
| $5^{7}$ | $1$ | $5$ | $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$ | |
| $5^{7}$ | $1$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ | |
| $2^{10},1^{15}$ | $105$ | $2$ | $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)(11,16)(12,17)(13,18)(14,19)(15,20)$ | |
| $10^{2},5^{3}$ | $105$ | $10$ | $( 1, 8, 5, 7, 4, 6, 3,10, 2, 9)(11,18,15,17,14,16,13,20,12,19)(21,23,25,22,24) (26,28,30,27,29)(31,33,35,32,34)$ | |
| $10^{2},5^{3}$ | $105$ | $10$ | $( 1,10, 4, 8, 2, 6, 5, 9, 3, 7)(11,20,14,18,12,16,15,19,13,17)(21,25,24,23,22) (26,30,29,28,27)(31,35,34,33,32)$ | |
| $10^{2},5^{3}$ | $105$ | $10$ | $( 1, 9, 2,10, 3, 6, 4, 7, 5, 8)(11,19,12,20,13,16,14,17,15,18)(21,24,22,25,23) (26,29,27,30,28)(31,34,32,35,33)$ | |
| $10^{2},5^{3}$ | $105$ | $10$ | $( 1, 7, 3, 9, 5, 6, 2, 8, 4,10)(11,17,13,19,15,16,12,18,14,20)(21,22,23,24,25) (26,27,28,29,30)(31,32,33,34,35)$ | |
| $3^{5},1^{20}$ | $70$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ | |
| $15,5^{4}$ | $70$ | $15$ | $( 1, 8,15, 2, 9,11, 3,10,12, 4, 6,13, 5, 7,14)(16,18,20,17,19)(21,23,25,22,24) (26,28,30,27,29)(31,33,35,32,34)$ | |
| $15,5^{4}$ | $70$ | $15$ | $( 1,10,14, 3, 7,11, 5, 9,13, 2, 6,15, 4, 8,12)(16,20,19,18,17)(21,25,24,23,22) (26,30,29,28,27)(31,35,34,33,32)$ | |
| $15,5^{4}$ | $70$ | $15$ | $( 1, 9,12, 5, 8,11, 4, 7,15, 3, 6,14, 2,10,13)(16,19,17,20,18)(21,24,22,25,23) (26,29,27,30,28)(31,34,32,35,33)$ | |
| $15,5^{4}$ | $70$ | $15$ | $( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)(16,17,18,19,20)(21,22,23,24,25) (26,27,28,29,30)(31,32,33,34,35)$ | |
| $3^{5},2^{10}$ | $210$ | $6$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)(16,21)(17,22)(18,23)(19,24) (20,25)(26,31)(27,32)(28,33)(29,34)(30,35)$ | |
| $15,10^{2}$ | $210$ | $30$ | $( 1, 8,15, 2, 9,11, 3,10,12, 4, 6,13, 5, 7,14)(16,23,20,22,19,21,18,25,17,24) (26,33,30,32,29,31,28,35,27,34)$ | |
| $15,10^{2}$ | $210$ | $30$ | $( 1,10,14, 3, 7,11, 5, 9,13, 2, 6,15, 4, 8,12)(16,25,19,23,17,21,20,24,18,22) (26,35,29,33,27,31,30,34,28,32)$ | |
| $15,10^{2}$ | $210$ | $30$ | $( 1, 9,12, 5, 8,11, 4, 7,15, 3, 6,14, 2,10,13)(16,24,17,25,18,21,19,22,20,23) (26,34,27,35,28,31,29,32,30,33)$ | |
| $15,10^{2}$ | $210$ | $30$ | $( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)(16,22,18,24,20,21,17,23,19,25) (26,32,28,34,30,31,27,33,29,35)$ | |
| $3^{10},1^{5}$ | $280$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)(16,21,26)(17,22,27) (18,23,28)(19,24,29)(20,25,30)$ | |
| $15^{2},5$ | $280$ | $15$ | $( 1, 8,15, 2, 9,11, 3,10,12, 4, 6,13, 5, 7,14)(16,23,30,17,24,26,18,25,27,19, 21,28,20,22,29)(31,33,35,32,34)$ | |
| $15^{2},5$ | $280$ | $15$ | $( 1,10,14, 3, 7,11, 5, 9,13, 2, 6,15, 4, 8,12)(16,25,29,18,22,26,20,24,28,17, 21,30,19,23,27)(31,35,34,33,32)$ | |
| $15^{2},5$ | $280$ | $15$ | $( 1, 9,12, 5, 8,11, 4, 7,15, 3, 6,14, 2,10,13)(16,24,27,20,23,26,19,22,30,18, 21,29,17,25,28)(31,34,32,35,33)$ | |
| $15^{2},5$ | $280$ | $15$ | $( 1, 7,13, 4,10,11, 2, 8,14, 5, 6,12, 3, 9,15)(16,22,28,19,25,26,17,23,29,20, 21,27,18,24,30)(31,32,33,34,35)$ | |
| $4^{5},2^{5},1^{5}$ | $630$ | $4$ | $( 1, 6,11,16)( 2, 7,12,17)( 3, 8,13,18)( 4, 9,14,19)( 5,10,15,20)(21,26) (22,27)(23,28)(24,29)(25,30)$ | |
| $20,10,5$ | $630$ | $20$ | $( 1, 8,15,17, 4, 6,13,20, 2, 9,11,18, 5, 7,14,16, 3,10,12,19)(21,28,25,27,24, 26,23,30,22,29)(31,33,35,32,34)$ | |
| $20,10,5$ | $630$ | $20$ | $( 1,10,14,18, 2, 6,15,19, 3, 7,11,20, 4, 8,12,16, 5, 9,13,17)(21,30,24,28,22, 26,25,29,23,27)(31,35,34,33,32)$ | |
| $20,10,5$ | $630$ | $20$ | $( 1, 9,12,20, 3, 6,14,17, 5, 8,11,19, 2,10,13,16, 4, 7,15,18)(21,29,22,30,23, 26,24,27,25,28)(31,34,32,35,33)$ | |
| $20,10,5$ | $630$ | $20$ | $( 1, 7,13,19, 5, 6,12,18, 4,10,11,17, 3, 9,15,16, 2, 8,14,20)(21,27,23,29,25, 26,22,28,24,30)(31,32,33,34,35)$ | |
| $5^{5},1^{10}$ | $504$ | $5$ | $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$ | |
| $5^{7}$ | $504$ | $5$ | $( 1, 8,15,17,24)( 2, 9,11,18,25)( 3,10,12,19,21)( 4, 6,13,20,22) ( 5, 7,14,16,23)(26,28,30,27,29)(31,33,35,32,34)$ | |
| $5^{7}$ | $504$ | $5$ | $( 1,10,14,18,22)( 2, 6,15,19,23)( 3, 7,11,20,24)( 4, 8,12,16,25) ( 5, 9,13,17,21)(26,30,29,28,27)(31,35,34,33,32)$ | |
| $5^{7}$ | $504$ | $5$ | $( 1, 9,12,20,23)( 2,10,13,16,24)( 3, 6,14,17,25)( 4, 7,15,18,21) ( 5, 8,11,19,22)(26,29,27,30,28)(31,34,32,35,33)$ | |
| $5^{7}$ | $504$ | $5$ | $( 1, 7,13,19,25)( 2, 8,14,20,21)( 3, 9,15,16,22)( 4,10,11,17,23) ( 5, 6,12,18,24)(26,27,28,29,30)(31,32,33,34,35)$ | |
| $7^{5}$ | $360$ | $7$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ | |
| $35$ | $360$ | $35$ | $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$ | |
| $35$ | $360$ | $35$ | $( 1,10,14,18,22,26,35, 4, 8,12,16,25,29,33, 2, 6,15,19,23,27,31, 5, 9,13,17, 21,30,34, 3, 7,11,20,24,28,32)$ | |
| $35$ | $360$ | $35$ | $( 1, 9,12,20,23,26,34, 2,10,13,16,24,27,35, 3, 6,14,17,25,28,31, 4, 7,15,18, 21,29,32, 5, 8,11,19,22,30,33)$ | |
| $35$ | $360$ | $35$ | $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$ | |
| $7^{5}$ | $360$ | $7$ | $( 1, 6,11,16,21,31,26)( 2, 7,12,17,22,32,27)( 3, 8,13,18,23,33,28) ( 4, 9,14,19,24,34,29)( 5,10,15,20,25,35,30)$ | |
| $35$ | $360$ | $35$ | $( 1, 8,15,17,24,31,28, 5, 7,14,16,23,35,27, 4, 6,13,20,22,34,26, 3,10,12,19, 21,33,30, 2, 9,11,18,25,32,29)$ | |
| $35$ | $360$ | $35$ | $( 1,10,14,18,22,31,30, 4, 8,12,16,25,34,28, 2, 6,15,19,23,32,26, 5, 9,13,17, 21,35,29, 3, 7,11,20,24,33,27)$ | |
| $35$ | $360$ | $35$ | $( 1, 9,12,20,23,31,29, 2,10,13,16,24,32,30, 3, 6,14,17,25,33,26, 4, 7,15,18, 21,34,27, 5, 8,11,19,22,35,28)$ | |
| $35$ | $360$ | $35$ | $( 1, 7,13,19,25,31,27, 3, 9,15,16,22,33,29, 5, 6,12,18,24,35,26, 2, 8,14,20, 21,32,28, 4,10,11,17,23,34,30)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $12600=2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7$ | 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: | 12600.r | magma: IdentifyGroup(G);
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| Character table: | 45 x 45 character table |
magma: CharacterTable(G);