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Properties

Label 35T27
Degree $35$
Order $2520$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $F_7\times A_5$

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Show commands: Magma

magma: G := TransitiveGroup(35, 27);
 

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_7\times A_5$
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:  (1,34,14,5,33,13,2,31,12,4,35,15,3,32,11)(6,16,22,9,20,21,8,18,23,7,19,25,10,17,24)(26,27,28,30,29), (1,12,27,35,22,9)(2,15,30,34,21,6)(3,14,29,31,25,8)(4,13,26,33,24,10)(5,11,28,32,23,7)(17,20,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$42$:  $F_7$
$60$:  $A_5$
$120$:  $A_5\times C_2$
$180$:  $\GL(2,4)$
$360$:  30T87

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $A_5$

Degree 7: $F_7$

Low degree siblings

42T297

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 7, 7, 7, 7, 7 $ $6$ $7$ $( 1,13,24,35, 7,20,28)( 2,15,21,34, 6,19,30)( 3,14,25,31, 8,16,29) ( 4,11,23,33, 9,17,27)( 5,12,22,32,10,18,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $7$ $2$ $( 6,34)( 7,35)( 8,31)( 9,33)(10,32)(11,27)(12,26)(13,28)(14,29)(15,30)(16,25) (17,23)(18,22)(19,21)(20,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $7$ $3$ $( 6,21,15)( 7,24,13)( 8,25,14)( 9,23,11)(10,22,12)(16,29,31)(17,27,33) (18,26,32)(19,30,34)(20,28,35)$
$ 6, 6, 6, 6, 6, 1, 1, 1, 1, 1 $ $7$ $6$ $( 6,19,15,34,21,30)( 7,20,13,35,24,28)( 8,16,14,31,25,29)( 9,17,11,33,23,27) (10,18,12,32,22,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $7$ $3$ $( 6,15,21)( 7,13,24)( 8,14,25)( 9,11,23)(10,12,22)(16,31,29)(17,33,27) (18,32,26)(19,34,30)(20,35,28)$
$ 6, 6, 6, 6, 6, 1, 1, 1, 1, 1 $ $7$ $6$ $( 6,30,21,34,15,19)( 7,28,24,35,13,20)( 8,29,25,31,14,16)( 9,27,23,33,11,17) (10,26,22,32,12,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1, 2)( 3, 4)( 6, 7)( 8, 9)(11,14)(13,15)(16,17)(19,20)(21,24)(23,25)(27,29) (28,30)(31,33)(34,35)$
$ 14, 14, 7 $ $90$ $14$ $( 1,15,24,34, 7,19,28, 2,13,21,35, 6,20,30)( 3,11,25,33, 8,17,29, 4,14,23,31, 9,16,27)( 5,12,22,32,10,18,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $105$ $2$ $( 1, 2)( 3, 4)( 6,35)( 7,34)( 8,33)( 9,31)(10,32)(11,29)(12,26)(13,30)(14,27) (15,28)(16,23)(17,25)(18,22)(19,24)(20,21)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $105$ $6$ $( 1, 2)( 3, 4)( 6,24,15, 7,21,13)( 8,23,14, 9,25,11)(10,22,12)(16,27,31,17,29, 33)(18,26,32)(19,28,34,20,30,35)$
$ 6, 6, 6, 6, 6, 2, 2, 1 $ $105$ $6$ $( 1, 2)( 3, 4)( 6,20,15,35,21,28)( 7,19,13,34,24,30)( 8,17,14,33,25,27) ( 9,16,11,31,23,29)(10,18,12,32,22,26)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $105$ $6$ $( 1, 2)( 3, 4)( 6,13,21, 7,15,24)( 8,11,25, 9,14,23)(10,12,22)(16,33,29,17,31, 27)(18,32,26)(19,35,30,20,34,28)$
$ 6, 6, 6, 6, 6, 2, 2, 1 $ $105$ $6$ $( 1, 2)( 3, 4)( 6,28,21,35,15,20)( 7,30,24,34,13,19)( 8,27,25,33,14,17) ( 9,29,23,31,11,16)(10,26,22,32,12,18)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 2, 3)( 6, 8, 7)(13,15,14)(16,20,19)(21,25,24)(28,30,29)(31,35,34)$
$ 21, 7, 7 $ $120$ $21$ $( 1,15,25,35, 6,16,28, 2,14,24,34, 8,20,30, 3,13,21,31, 7,19,29) ( 4,11,23,33, 9,17,27)( 5,12,22,32,10,18,26)$
$ 6, 6, 6, 3, 2, 2, 2, 2, 2, 2, 1, 1 $ $140$ $6$ $( 1, 2, 3)( 6,31, 7,34, 8,35)( 9,33)(10,32)(11,27)(12,26)(13,30,14,28,15,29) (16,24,19,25,20,21)(17,23)(18,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $140$ $3$ $( 1, 2, 3)( 6,25,13)( 7,21,14)( 8,24,15)( 9,23,11)(10,22,12)(16,28,34) (17,27,33)(18,26,32)(19,29,35)(20,30,31)$
$ 6, 6, 6, 6, 6, 3, 1, 1 $ $140$ $6$ $( 1, 2, 3)( 6,16,13,34,25,28)( 7,19,14,35,21,29)( 8,20,15,31,24,30) ( 9,17,11,33,23,27)(10,18,12,32,22,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $140$ $3$ $( 1, 2, 3)( 6,14,24)( 7,15,25)( 8,13,21)( 9,11,23)(10,12,22)(16,35,30) (17,33,27)(18,32,26)(19,31,28)(20,34,29)$
$ 6, 6, 6, 6, 6, 3, 1, 1 $ $140$ $6$ $( 1, 2, 3)( 6,29,24,34,14,20)( 7,30,25,35,15,16)( 8,28,21,31,13,19) ( 9,27,23,33,11,17)(10,26,22,32,12,18)$
$ 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 2, 3, 4, 5)( 6, 8, 9,10, 7)(11,12,13,15,14)(16,17,18,20,19) (21,25,23,22,24)(26,28,30,29,27)(31,33,32,35,34)$
$ 35 $ $72$ $35$ $( 1,15,25,33,10,20,30, 3,11,22,35, 6,16,27, 5,13,21,31, 9,18,28, 2,14,23,32, 7,19,29, 4,12,24,34, 8,17,26)$
$ 10, 10, 10, 5 $ $84$ $10$ $( 1, 2, 3, 4, 5)( 6,31, 9,32, 7,34, 8,33,10,35)(11,26,13,30,14,27,12,28,15,29) (16,23,18,24,19,25,17,22,20,21)$
$ 15, 15, 5 $ $84$ $15$ $( 1, 2, 3, 4, 5)( 6,25,11,10,24,15, 8,23,12, 7,21,14, 9,22,13)(16,27,32,20,30, 31,17,26,35,19,29,33,18,28,34)$
$ 30, 5 $ $84$ $30$ $( 1, 2, 3, 4, 5)( 6,16,11,32,24,30, 8,17,12,35,21,29, 9,18,13,34,25,27,10,20, 15,31,23,26, 7,19,14,33,22,28)$
$ 15, 15, 5 $ $84$ $15$ $( 1, 2, 3, 4, 5)( 6,14,23,10,13,21, 8,11,22, 7,15,25, 9,12,24)(16,33,26,20,34, 29,17,32,28,19,31,27,18,35,30)$
$ 30, 5 $ $84$ $30$ $( 1, 2, 3, 4, 5)( 6,29,23,32,13,19, 8,27,22,35,15,16, 9,26,24,34,14,17,10,28, 21,31,11,18, 7,30,25,33,12,20)$
$ 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 2, 3, 5, 4)( 6, 8,10, 9, 7)(11,13,15,14,12)(16,18,17,20,19) (21,25,22,23,24)(26,27,28,30,29)(31,32,33,35,34)$
$ 35 $ $72$ $35$ $( 1,15,25,32, 9,20,30, 3,12,23,35, 6,16,26, 4,13,21,31,10,17,28, 2,14,22,33, 7,19,29, 5,11,24,34, 8,18,27)$
$ 10, 10, 10, 5 $ $84$ $10$ $( 1, 2, 3, 5, 4)( 6,31,10,33, 7,34, 8,32, 9,35)(11,28,15,29,12,27,13,30,14,26) (16,22,17,24,19,25,18,23,20,21)$
$ 15, 15, 5 $ $84$ $15$ $( 1, 2, 3, 5, 4)( 6,25,12, 9,24,15, 8,22,11, 7,21,14,10,23,13)(16,26,33,20,30, 31,18,27,35,19,29,32,17,28,34)$
$ 30, 5 $ $84$ $30$ $( 1, 2, 3, 5, 4)( 6,16,12,33,24,30, 8,18,11,35,21,29,10,17,13,34,25,26, 9,20, 15,31,22,27, 7,19,14,32,23,28)$
$ 15, 15, 5 $ $84$ $15$ $( 1, 2, 3, 5, 4)( 6,14,22, 9,13,21, 8,12,23, 7,15,25,10,11,24)(16,32,27,20,34, 29,18,33,28,19,31,26,17,35,30)$
$ 30, 5 $ $84$ $30$ $( 1, 2, 3, 5, 4)( 6,29,22,33,13,19, 8,26,23,35,15,16,10,27,24,34,14,18, 9,28, 21,31,12,17, 7,30,25,32,11,20)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  2520.bl
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);