Show commands:
Magma
magma: G := TransitiveGroup(35, 29);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $29$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $F_5\times \GL(3,2)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,30,32,23,6,20,12,3,26,35,22,8,16,15,2,28,31,25,7,18,11,5,27,33,21,10,17,13)(4,29,34,24,9,19,14), (1,7,35,24)(2,10,34,21)(3,8,33,23)(4,6,32,25)(5,9,31,22)(11,12,15,14)(16,27,20,29)(17,30,19,26)(18,28) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $20$: $F_5$ $168$: $\GL(3,2)$ $336$: 14T17 $672$: 28T86 Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Degree 7: $\GL(3,2)$
Low degree siblings
35T29, 40T2719Siblings 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 |
| $1^{35}$ | $1$ | $1$ | $()$ | |
| $5^{7}$ | $4$ | $5$ | $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$ | |
| $2^{14},1^{7}$ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ | |
| $4^{7},1^{7}$ | $5$ | $4$ | $( 2, 3, 5, 4)( 7, 8,10, 9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29) (32,33,35,34)$ | |
| $4^{7},1^{7}$ | $5$ | $4$ | $( 2, 4, 5, 3)( 7, 9,10, 8)(12,14,15,13)(17,19,20,18)(22,24,25,23)(27,29,30,28) (32,34,35,33)$ | |
| $2^{16},1^{3}$ | $105$ | $2$ | $( 2, 5)( 3, 4)( 6,31)( 7,35)( 8,34)( 9,33)(10,32)(11,16)(12,20)(13,19)(14,18) (15,17)(22,25)(23,24)(27,30)(28,29)$ | |
| $2^{10},1^{15}$ | $21$ | $2$ | $( 6,31)( 7,32)( 8,33)( 9,34)(10,35)(11,16)(12,17)(13,18)(14,19)(15,20)$ | |
| $10^{2},5^{3}$ | $84$ | $10$ | $( 1, 5, 4, 3, 2)( 6,35, 9,33, 7,31,10,34, 8,32)(11,20,14,18,12,16,15,19,13,17) (21,25,24,23,22)(26,30,29,28,27)$ | |
| $4^{7},2^{2},1^{3}$ | $105$ | $4$ | $( 2, 4, 5, 3)( 6,31)( 7,34,10,33)( 8,32, 9,35)(11,16)(12,19,15,18) (13,17,14,20)(22,24,25,23)(27,29,30,28)$ | |
| $4^{7},2^{2},1^{3}$ | $105$ | $4$ | $( 2, 3, 5, 4)( 6,31)( 7,33,10,34)( 8,35, 9,32)(11,16)(12,18,15,19) (13,20,14,17)(22,23,25,24)(27,28,30,29)$ | |
| $12^{2},4,3^{2},1$ | $280$ | $12$ | $( 1, 3, 4, 2)( 6,33,29, 7,31,28, 9,32,26, 8,34,27)(10,35,30)(11,23,19,12,21, 18,14,22,16,13,24,17)(15,25,20)$ | |
| $12^{2},4,3^{2},1$ | $280$ | $12$ | $( 1, 4, 5, 2)( 6,34,30, 7,31,29,10,32,26, 9,35,27)( 8,33,28)(11,24,20,12,21, 19,15,22,16,14,25,17)(13,23,18)$ | |
| $15^{2},5$ | $224$ | $15$ | $( 1, 5, 4, 3, 2)( 6,35,29, 8,32,26,10,34,28, 7,31,30, 9,33,27)(11,25,19,13,22, 16,15,24,18,12,21,20,14,23,17)$ | |
| $3^{10},1^{5}$ | $56$ | $3$ | $( 6,31,26)( 7,32,27)( 8,33,28)( 9,34,29)(10,35,30)(11,21,16)(12,22,17) (13,23,18)(14,24,19)(15,25,20)$ | |
| $6^{4},3^{2},2^{2},1$ | $280$ | $6$ | $( 1, 2)( 3, 5)( 6,32,26, 7,31,27)( 8,35,28,10,33,30)( 9,34,29)(11,22,16,12,21, 17)(13,25,18,15,23,20)(14,24,19)$ | |
| $4^{8},2,1$ | $210$ | $4$ | $( 1, 5, 3, 4)( 6,25,18,29)( 7,22,17,27)( 8,24,16,30)( 9,21,20,28)(10,23,19,26) (11,35,13,34)(12,32)(14,31,15,33)$ | |
| $4^{8},2,1$ | $210$ | $4$ | $( 1, 2, 5, 4)( 6,22,20,29)( 7,25,19,26)( 8,23,18,28)( 9,21,17,30)(10,24,16,27) (11,32,15,34)(12,35,14,31)(13,33)$ | |
| $4^{5},2^{7},1$ | $210$ | $4$ | $( 1, 4)( 2, 3)( 6,24,16,29)( 7,23,17,28)( 8,22,18,27)( 9,21,19,26) (10,25,20,30)(11,34)(12,33)(13,32)(14,31)(15,35)$ | |
| $20,10,5$ | $168$ | $20$ | $( 1, 3, 5, 2, 4)( 6,23,20,27, 9,21,18,30, 7,24,16,28,10,22,19,26, 8,25,17,29) (11,33,15,32,14,31,13,35,12,34)$ | |
| $4^{5},2^{5},1^{5}$ | $42$ | $4$ | $( 6,21,16,26)( 7,22,17,27)( 8,23,18,28)( 9,24,19,29)(10,25,20,30)(11,31) (12,32)(13,33)(14,34)(15,35)$ | |
| $35$ | $96$ | $35$ | $( 1,28,10,17,14,21,33, 5,27, 9,16,13,25,32, 4,26, 8,20,12,24,31, 3,30, 7,19, 11,23,35, 2,29, 6,18,15,22,34)$ | |
| $7^{5}$ | $24$ | $7$ | $( 1,26, 6,16,11,21,31)( 2,27, 7,17,12,22,32)( 3,28, 8,18,13,23,33) ( 4,29, 9,19,14,24,34)( 5,30,10,20,15,25,35)$ | |
| $14^{2},7$ | $120$ | $14$ | $( 1,29, 6,19,11,24,31, 4,26, 9,16,14,21,34)( 2,28, 7,18,12,23,32, 3,27, 8,17, 13,22,33)( 5,30,10,20,15,25,35)$ | |
| $28,7$ | $120$ | $28$ | $( 1,30, 8,19,11,25,33, 4,26,10,18,14,21,35, 3,29, 6,20,13,24,31, 5,28, 9,16, 15,23,34)( 2,27, 7,17,12,22,32)$ | |
| $28,7$ | $120$ | $28$ | $( 1,27,10,19,11,22,35, 4,26, 7,20,14,21,32, 5,29, 6,17,15,24,31, 2,30, 9,16, 12,25,34)( 3,28, 8,18,13,23,33)$ | |
| $14^{2},7$ | $120$ | $14$ | $( 1,27, 6,22,31,12,16, 2,26, 7,21,32,11,17)( 3,30, 8,25,33,15,18, 5,28,10,23, 35,13,20)( 4,29, 9,24,34,14,19)$ | |
| $35$ | $96$ | $35$ | $( 1,30, 9,23,32,11,20, 4,28, 7,21,35,14,18, 2,26,10,24,33,12,16, 5,29, 8,22, 31,15,19, 3,27, 6,25,34,13,17)$ | |
| $7^{5}$ | $24$ | $7$ | $( 1,26, 6,21,31,11,16)( 2,27, 7,22,32,12,17)( 3,28, 8,23,33,13,18) ( 4,29, 9,24,34,14,19)( 5,30,10,25,35,15,20)$ | |
| $28,7$ | $120$ | $28$ | $( 1,28, 9,22,31,13,19, 2,26, 8,24,32,11,18, 4,27, 6,23,34,12,16, 3,29, 7,21, 33,14,17)( 5,30,10,25,35,15,20)$ | |
| $28,7$ | $120$ | $28$ | $( 1,29,10,22,31,14,20, 2,26, 9,25,32,11,19, 5,27, 6,24,35,12,16, 4,30, 7,21, 34,15,17)( 3,28, 8,23,33,13,18)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $3360=2^{5} \cdot 3 \cdot 5 \cdot 7$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | no | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 3360.v | magma: IdentifyGroup(G);
| |
| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 7 P | |
| Type |
magma: CharacterTable(G);