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Magma
magma: G := TransitiveGroup(35, 35);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_5\times \GL(3,2)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,21,11,30,10,19,35,5,25,15,26,8,18,34,2,22,14,28,9,16,33,3,24,12,27,6,20,32)(4,23,13,29,7,17,31), (1,29,35)(2,27,31)(3,30,34,5,28,32)(4,26,33)(6,12,21,8,15,22)(7,11,25)(9,14,23)(10,13,24)(16,19)(17,20,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $120$: $S_5$ $168$: $\GL(3,2)$ $336$: 14T17 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $S_5$
Degree 7: $\GL(3,2)$
Low degree siblings
35T35, 40T11250, 42T723 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has exactly one arithmetically equivalent field.
Conjugacy classes
| Label | Cycle Type | Size | Order | Representative |
| $1^{35}$ | $1$ | $1$ | $()$ | |
| $2^{10},1^{15}$ | $21$ | $2$ | $( 6,28)( 7,29)( 8,30)( 9,26)(10,27)(16,22)(17,23)(18,25)(19,21)(20,24)$ | |
| $3^{10},1^{5}$ | $56$ | $3$ | $( 6,28,15)( 7,29,13)( 8,30,12)( 9,26,11)(10,27,14)(16,22,32)(17,23,31) (18,25,33)(19,21,34)(20,24,35)$ | |
| $4^{5},2^{5},1^{5}$ | $42$ | $4$ | $( 6,34,19,15)( 7,31,17,13)( 8,32,16,12)( 9,35,20,11)(10,33,18,14)(21,28) (22,30)(23,29)(24,26)(25,27)$ | |
| $7^{5}$ | $24$ | $7$ | $( 1,33,18,14,25,27,10)( 2,35,20,11,24,26, 9)( 3,34,19,15,21,28, 6) ( 4,31,17,13,23,29, 7)( 5,32,16,12,22,30, 8)$ | |
| $7^{5}$ | $24$ | $7$ | $( 1,33,18,10,27,14,25)( 2,35,20, 9,26,11,24)( 3,34,19, 6,28,15,21) ( 4,31,17, 7,29,13,23)( 5,32,16, 8,30,12,22)$ | |
| $2^{7},1^{21}$ | $10$ | $2$ | $( 1, 2)( 9,10)(11,14)(18,20)(24,25)(26,27)(33,35)$ | |
| $2^{13},1^{9}$ | $210$ | $2$ | $( 1, 2)( 6,28)( 7,29)( 8,30)( 9,27)(10,26)(11,14)(16,22)(17,23)(18,24)(19,21) (20,25)(33,35)$ | |
| $6^{2},3^{6},2,1^{3}$ | $560$ | $6$ | $( 1, 2)( 6,28,15)( 7,29,13)( 8,30,12)( 9,27,11,10,26,14)(16,22,32)(17,23,31) (18,24,33,20,25,35)(19,21,34)$ | |
| $4^{5},2^{6},1^{3}$ | $420$ | $4$ | $( 1, 2)( 6,34,19,15)( 7,31,17,13)( 8,32,16,12)( 9,33,20,14)(10,35,18,11) (21,28)(22,30)(23,29)(24,27)(25,26)$ | |
| $14,7^{3}$ | $240$ | $14$ | $( 1,35,18,11,25,26,10, 2,33,20,14,24,27, 9)( 3,34,19,15,21,28, 6) ( 4,31,17,13,23,29, 7)( 5,32,16,12,22,30, 8)$ | |
| $14,7^{3}$ | $240$ | $14$ | $( 1,35,18, 9,27,11,25, 2,33,20,10,26,14,24)( 3,34,19, 6,28,15,21) ( 4,31,17, 7,29,13,23)( 5,32,16, 8,30,12,22)$ | |
| $2^{14},1^{7}$ | $15$ | $2$ | $( 1, 2)( 3, 4)( 6, 7)( 9,10)(11,14)(13,15)(17,19)(18,20)(21,23)(24,25)(26,27) (28,29)(31,34)(33,35)$ | |
| $2^{16},1^{3}$ | $315$ | $2$ | $( 1, 2)( 3, 4)( 6,29)( 7,28)( 8,30)( 9,27)(10,26)(11,14)(13,15)(16,22)(17,21) (18,24)(19,23)(20,25)(31,34)(33,35)$ | |
| $6^{4},3^{2},2^{2},1$ | $840$ | $6$ | $( 1, 2)( 3, 4)( 6,29,15, 7,28,13)( 8,30,12)( 9,27,11,10,26,14)(16,22,32) (17,21,31,19,23,34)(18,24,33,20,25,35)$ | |
| $4^{5},2^{7},1$ | $630$ | $4$ | $( 1, 2)( 3, 4)( 6,31,19,13)( 7,34,17,15)( 8,32,16,12)( 9,33,20,14) (10,35,18,11)(21,29)(22,30)(23,28)(24,27)(25,26)$ | |
| $14^{2},7$ | $360$ | $14$ | $( 1,35,18,11,25,26,10, 2,33,20,14,24,27, 9)( 3,31,19,13,21,29, 6, 4,34,17,15, 23,28, 7)( 5,32,16,12,22,30, 8)$ | |
| $14^{2},7$ | $360$ | $14$ | $( 1,35,18, 9,27,11,25, 2,33,20,10,26,14,24)( 3,31,19, 7,28,13,21, 4,34,17, 6, 29,15,23)( 5,32,16, 8,30,12,22)$ | |
| $3^{7},1^{14}$ | $20$ | $3$ | $( 1, 2, 3)( 6,10, 9)(11,15,14)(18,20,19)(21,25,24)(26,28,27)(33,35,34)$ | |
| $6^{2},3^{3},2^{4},1^{6}$ | $420$ | $6$ | $( 1, 2, 3)( 6,27, 9,28,10,26)( 7,29)( 8,30)(11,15,14)(16,22)(17,23) (18,24,19,25,20,21)(33,35,34)$ | |
| $3^{11},1^{2}$ | $1120$ | $3$ | $( 1, 2, 3)( 6,27,11)( 7,29,13)( 8,30,12)( 9,28,14)(10,26,15)(16,22,32) (17,23,31)(18,24,34)(19,25,35)(20,21,33)$ | |
| $12,6,4^{2},3,2^{2},1^{2}$ | $840$ | $12$ | $( 1, 2, 3)( 6,33,20,15,10,35,19,14, 9,34,18,11)( 7,31,17,13)( 8,32,16,12) (21,27,24,28,25,26)(22,30)(23,29)$ | |
| $21,7^{2}$ | $480$ | $21$ | $( 1,35,19,14,24,28,10, 2,34,18,11,21,27, 9, 3,33,20,15,25,26, 6) ( 4,31,17,13,23,29, 7)( 5,32,16,12,22,30, 8)$ | |
| $21,7^{2}$ | $480$ | $21$ | $( 1,35,19,10,26,15,25, 2,34,18, 9,28,14,24, 3,33,20, 6,27,11,21) ( 4,31,17, 7,29,13,23)( 5,32,16, 8,30,12,22)$ | |
| $3^{7},2^{7}$ | $20$ | $6$ | $( 1, 2, 3)( 4, 5)( 6,10, 9)( 7, 8)(11,15,14)(12,13)(16,17)(18,20,19)(21,25,24) (22,23)(26,28,27)(29,30)(31,32)(33,35,34)$ | |
| $6^{2},3^{3},2^{7}$ | $420$ | $6$ | $( 1, 2, 3)( 4, 5)( 6,27, 9,28,10,26)( 7,30)( 8,29)(11,15,14)(12,13)(16,23) (17,22)(18,24,19,25,20,21)(31,32)(33,35,34)$ | |
| $6^{2},3^{7},2$ | $1120$ | $6$ | $( 1, 2, 3)( 4, 5)( 6,27,11)( 7,30,13, 8,29,12)( 9,28,14)(10,26,15) (16,23,32,17,22,31)(18,24,34)(19,25,35)(20,21,33)$ | |
| $12,6,4^{2},3,2^{3}$ | $840$ | $12$ | $( 1, 2, 3)( 4, 5)( 6,33,20,15,10,35,19,14, 9,34,18,11)( 7,32,17,12) ( 8,31,16,13)(21,27,24,28,25,26)(22,29)(23,30)$ | |
| $21,14$ | $480$ | $42$ | $( 1,35,19,14,24,28,10, 2,34,18,11,21,27, 9, 3,33,20,15,25,26, 6) ( 4,32,17,12,23,30, 7, 5,31,16,13,22,29, 8)$ | |
| $21,14$ | $480$ | $42$ | $( 1,35,19,10,26,15,25, 2,34,18, 9,28,14,24, 3,33,20, 6,27,11,21) ( 4,32,17, 8,29,12,23, 5,31,16, 7,30,13,22)$ | |
| $4^{7},1^{7}$ | $30$ | $4$ | $( 1, 2, 3, 4)( 6, 7,10, 9)(11,15,13,14)(17,18,20,19)(21,23,25,24)(26,28,29,27) (31,33,35,34)$ | |
| $4^{7},2^{2},1^{3}$ | $630$ | $4$ | $( 1, 2, 3, 4)( 6,29,10,26)( 7,27, 9,28)( 8,30)(11,15,13,14)(16,22) (17,25,20,21)(18,24,19,23)(31,33,35,34)$ | |
| $12^{2},4,3^{2},1$ | $1680$ | $12$ | $( 1, 2, 3, 4)( 6,29,14, 9,28,13,10,26,15, 7,27,11)( 8,30,12)(16,22,32) (17,25,35,19,23,33,20,21,31,18,24,34)$ | |
| $4^{8},2,1$ | $1260$ | $4$ | $( 1, 2, 3, 4)( 6,31,18,11)( 7,33,20,15)( 8,32,16,12)( 9,34,17,14)(10,35,19,13) (21,29,25,26)(22,30)(23,27,24,28)$ | |
| $28,7$ | $720$ | $28$ | $( 1,35,19,13,25,26, 6, 4,33,20,15,23,27, 9, 3,31,18,11,21,29,10, 2,34,17,14, 24,28, 7)( 5,32,16,12,22,30, 8)$ | |
| $28,7$ | $720$ | $28$ | $( 1,35,19, 7,27,11,21, 4,33,20, 6,29,14,24, 3,31,18, 9,28,13,25, 2,34,17,10, 26,15,23)( 5,32,16, 8,30,12,22)$ | |
| $5^{7}$ | $24$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8,10, 9)(11,15,13,12,14)(16,18,20,19,17) (21,23,22,25,24)(26,28,29,30,27)(31,32,33,35,34)$ | |
| $10^{2},5^{3}$ | $504$ | $10$ | $( 1, 2, 3, 4, 5)( 6,29, 8,27, 9,28, 7,30,10,26)(11,15,13,12,14) (16,25,20,21,17,22,18,24,19,23)(31,32,33,35,34)$ | |
| $15^{2},5$ | $1344$ | $15$ | $( 1, 2, 3, 4, 5)( 6,29,12,10,26,15, 7,30,14, 9,28,13, 8,27,11)(16,25,35,19,23, 32,18,24,34,17,22,33,20,21,31)$ | |
| $20,10,5$ | $1008$ | $20$ | $( 1, 2, 3, 4, 5)( 6,31,16,14, 9,34,17,12,10,35,19,13, 8,33,20,15, 7,32,18,11) (21,29,22,27,24,28,23,30,25,26)$ | |
| $35$ | $576$ | $35$ | $( 1,35,19,13,22,27, 9, 3,31,16,14,24,28, 7, 5,33,20,15,23,30,10, 2,34,17,12, 25,26, 6, 4,32,18,11,21,29, 8)$ | |
| $35$ | $576$ | $35$ | $( 1,35,19, 7,30,14,24, 3,31,16,10,26,15,23, 5,33,20, 6,29,12,25, 2,34,17, 8, 27,11,21, 4,32,18, 9,28,13,22)$ |
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.n | magma: IdentifyGroup(G);
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| Character table: | 42 x 42 character table |
magma: CharacterTable(G);