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Properties

Label 35T32
Degree $35$
Order $10080$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $A_5\times \GL(3,2)$

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Show commands: Magma

magma: G := TransitiveGroup(35, 32);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$60$:  $A_5$
$168$:  $\GL(3,2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $A_5$

Degree 7: $\GL(3,2)$

Low degree siblings

35T32, 40T5863, 42T552 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 6,14)( 7,12)( 8,11)( 9,15)(10,13)(16,34)(17,32)(18,31)(19,35)(20,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $56$ $3$ $( 6,30,14)( 7,27,12)( 8,28,11)( 9,26,15)(10,29,13)(16,23,34)(17,24,32) (18,25,31)(19,22,35)(20,21,33)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $42$ $4$ $( 6,16)( 7,18)( 8,17)( 9,19)(10,20)(11,24,32,28)(12,25,31,27)(13,21,33,29) (14,23,34,30)(15,22,35,26)$
$ 7, 7, 7, 7, 7 $ $24$ $7$ $( 1,26,15,35, 9,22,19)( 2,29,13,33,10,21,20)( 3,30,14,34, 6,23,16) ( 4,28,11,32, 8,24,17)( 5,27,12,31, 7,25,18)$
$ 7, 7, 7, 7, 7 $ $24$ $7$ $( 1,26,15,22,19, 9,35)( 2,29,13,21,20,10,33)( 3,30,14,23,16, 6,34) ( 4,28,11,24,17, 8,32)( 5,27,12,25,18, 7,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1, 2)( 3, 4)( 6, 8)( 9,10)(11,14)(13,15)(16,17)(19,20)(21,22)(23,24)(26,29) (28,30)(32,34)(33,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $315$ $2$ $( 1, 2)( 3, 4)( 6,11)( 7,12)( 8,14)( 9,13)(10,15)(16,32)(17,34)(18,31)(19,33) (20,35)(21,22)(23,24)(26,29)(28,30)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $840$ $6$ $( 1, 2)( 3, 4)( 6,28,14, 8,30,11)( 7,27,12)( 9,29,15,10,26,13)(16,24,34,17,23, 32)(18,25,31)(19,21,35,20,22,33)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ $630$ $4$ $( 1, 2)( 3, 4)( 6,17)( 7,18)( 8,16)( 9,20)(10,19)(11,23,32,30)(12,25,31,27) (13,22,33,26)(14,24,34,28)(15,21,35,29)$
$ 14, 14, 7 $ $360$ $14$ $( 1,29,15,33, 9,21,19, 2,26,13,35,10,22,20)( 3,28,14,32, 6,24,16, 4,30,11,34, 8,23,17)( 5,27,12,31, 7,25,18)$
$ 14, 14, 7 $ $360$ $14$ $( 1,29,15,21,19,10,35, 2,26,13,22,20, 9,33)( 3,28,14,24,16, 8,34, 4,30,11,23, 17, 6,32)( 5,27,12,25,18, 7,31)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 2, 3)( 6, 9,10)(13,14,15)(16,19,20)(21,23,22)(26,29,30)(33,34,35)$
$ 6, 6, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $420$ $6$ $( 1, 2, 3)( 6,15,10,14, 9,13)( 7,12)( 8,11)(16,35,20,34,19,33)(17,32)(18,31) (21,23,22)(26,29,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $1120$ $3$ $( 1, 2, 3)( 6,26,13)( 7,27,12)( 8,28,11)( 9,29,14)(10,30,15)(16,22,33) (17,24,32)(18,25,31)(19,21,34)(20,23,35)$
$ 12, 6, 4, 4, 3, 2, 2, 1, 1 $ $840$ $12$ $( 1, 2, 3)( 6,19,10,16, 9,20)( 7,18)( 8,17)(11,24,32,28)(12,25,31,27) (13,23,35,29,14,22,33,30,15,21,34,26)$
$ 21, 7, 7 $ $480$ $21$ $( 1,29,14,35,10,23,19, 2,30,15,33, 6,22,20, 3,26,13,34, 9,21,16) ( 4,28,11,32, 8,24,17)( 5,27,12,31, 7,25,18)$
$ 21, 7, 7 $ $480$ $21$ $( 1,29,14,22,20, 6,35, 2,30,15,21,16, 9,33, 3,26,13,23,19,10,34) ( 4,28,11,24,17, 8,32)( 5,27,12,25,18, 7,31)$
$ 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 2, 3, 4, 5)( 6, 8, 7, 9,10)(11,12,15,13,14)(16,17,18,19,20) (21,23,24,25,22)(26,29,30,28,27)(31,35,33,34,32)$
$ 10, 10, 5, 5, 5 $ $252$ $10$ $( 1, 2, 3, 4, 5)( 6,11, 7,15,10,14, 8,12, 9,13)(16,32,18,35,20,34,17,31,19,33) (21,23,24,25,22)(26,29,30,28,27)$
$ 15, 15, 5 $ $672$ $15$ $( 1, 2, 3, 4, 5)( 6,28,12, 9,29,14, 8,27,15,10,30,11, 7,26,13)(16,24,31,19,21, 34,17,25,35,20,23,32,18,22,33)$
$ 20, 10, 5 $ $504$ $20$ $( 1, 2, 3, 4, 5)( 6,17, 7,19,10,16, 8,18, 9,20)(11,25,35,29,14,24,31,26,13,23, 32,27,15,21,34,28,12,22,33,30)$
$ 35 $ $288$ $35$ $( 1,29,14,32, 7,22,20, 3,28,12,35,10,23,17, 5,26,13,34, 8,25,19, 2,30,11,31, 9,21,16, 4,27,15,33, 6,24,18)$
$ 35 $ $288$ $35$ $( 1,29,14,24,18, 9,33, 3,28,12,22,20, 6,32, 5,26,13,23,17, 7,35, 2,30,11,25, 19,10,34, 4,27,15,21,16, 8,31)$
$ 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 2, 3, 5, 4)( 6, 7, 8, 9,10)(11,15,13,14,12)(16,18,17,19,20) (21,23,25,24,22)(26,29,30,27,28)(31,32,35,33,34)$
$ 10, 10, 5, 5, 5 $ $252$ $10$ $( 1, 2, 3, 5, 4)( 6,12, 8,15,10,14, 7,11, 9,13)(16,31,17,35,20,34,18,32,19,33) (21,23,25,24,22)(26,29,30,27,28)$
$ 15, 15, 5 $ $672$ $15$ $( 1, 2, 3, 5, 4)( 6,27,11, 9,29,14, 7,28,15,10,30,12, 8,26,13)(16,25,32,19,21, 34,18,24,35,20,23,31,17,22,33)$
$ 20, 10, 5 $ $504$ $20$ $( 1, 2, 3, 5, 4)( 6,18, 8,19,10,16, 7,17, 9,20)(11,22,33,30,12,24,35,29,14,25, 32,26,13,23,31,28,15,21,34,27)$
$ 35 $ $288$ $35$ $( 1,29,14,31, 8,22,20, 3,27,11,35,10,23,18, 4,26,13,34, 7,24,19, 2,30,12,32, 9,21,16, 5,28,15,33, 6,25,17)$
$ 35 $ $288$ $35$ $( 1,29,14,25,17, 9,33, 3,27,11,22,20, 6,31, 4,26,13,23,18, 8,35, 2,30,12,24, 19,10,34, 5,28,15,21,16, 7,32)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $10080=2^{5} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  10080.p
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);