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Properties

Label 26T48
Degree $26$
Order $11232$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\GL(3,3)$

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Show commands: Magma

magma: G := TransitiveGroup(26, 48);
 

Group action invariants

Degree $n$:  $26$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $48$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\GL(3,3)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2)(3,23,19,22,8,14)(4,24,20,21,7,13)(5,6)(9,12,16,10,11,15)(17,25)(18,26), (1,15,24,12,25,6,19,17,13,10,3,7,21,2,16,23,11,26,5,20,18,14,9,4,8,22)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5616$:  $\PSL(3,3)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: $\PSL(3,3)$

Low degree siblings

26T47 x 2, 26T48

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{26}$ $1$ $1$ $()$
$2^{13}$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)$
$2^{8},1^{10}$ $117$ $2$ $( 1, 8)( 2, 7)(11,25)(12,26)(13,16)(14,15)(17,23)(18,24)$
$2^{13}$ $117$ $2$ $( 1, 7)( 2, 8)( 3, 4)( 5, 6)( 9,10)(11,26)(12,25)(13,15)(14,16)(17,24)(18,23) (19,20)(21,22)$
$4^{4},2^{4},1^{2}$ $702$ $4$ $( 1,25, 8,11)( 2,26, 7,12)( 3,21)( 4,22)( 5, 9)( 6,10)(13,24,16,18) (14,23,15,17)$
$4^{4},2^{5}$ $702$ $4$ $( 1,26, 8,12)( 2,25, 7,11)( 3,22)( 4,21)( 5,10)( 6, 9)(13,23,16,17) (14,24,15,18)(19,20)$
$8^{2},4^{2},1^{2}$ $702$ $8$ $( 1,13,25,24, 8,16,11,18)( 2,14,26,23, 7,15,12,17)( 3, 5,21, 9)( 4, 6,22,10)$
$8^{2},4^{2},2$ $702$ $8$ $( 1,14,25,23, 8,15,11,17)( 2,13,26,24, 7,16,12,18)( 3, 6,21,10)( 4, 5,22, 9) (19,20)$
$8^{2},4^{2},1^{2}$ $702$ $8$ $( 1,18,11,16, 8,24,25,13)( 2,17,12,15, 7,23,26,14)( 3, 9,21, 5)( 4,10,22, 6)$
$8^{2},4^{2},2$ $702$ $8$ $( 1,17,11,15, 8,23,25,14)( 2,18,12,16, 7,24,26,13)( 3,10,21, 6)( 4, 9,22, 5) (19,20)$
$3^{6},1^{8}$ $104$ $3$ $( 1, 8,19)( 2, 7,20)( 5,21, 9)( 6,22,10)(15,23,26)(16,24,25)$
$6^{3},2^{4}$ $104$ $6$ $( 1, 7,19, 2, 8,20)( 3, 4)( 5,22, 9, 6,21,10)(11,12)(13,14)(15,24,26,16,23,25) (17,18)$
$6^{2},3^{2},2^{2},1^{4}$ $936$ $6$ $( 1,19, 8)( 2,20, 7)( 5,24,21,25, 9,16)( 6,23,22,26,10,15)(11,13)(12,14)$
$6^{3},2^{4}$ $936$ $6$ $( 1,20, 8, 2,19, 7)( 3, 4)( 5,23,21,26, 9,15)( 6,24,22,25,10,16)(11,14)(12,13) (17,18)$
$13^{2}$ $432$ $13$ $( 1,18, 8, 9,11, 3,19,24, 5,21,25,16,13)( 2,17, 7,10,12, 4,20,23, 6,22,26,15, 14)$
$26$ $432$ $26$ $( 1,17, 8,10,11, 4,19,23, 5,22,25,15,13, 2,18, 7, 9,12, 3,20,24, 6,21,26,16,14 )$
$13^{2}$ $432$ $13$ $( 1, 5, 9,16,19,18,21,11,13,24, 8,25, 3)( 2, 6,10,15,20,17,22,12,14,23, 7,26, 4)$
$26$ $432$ $26$ $( 1, 6, 9,15,19,17,21,12,13,23, 8,26, 3, 2, 5,10,16,20,18,22,11,14,24, 7,25, 4 )$
$13^{2}$ $432$ $13$ $( 1,13,16,25,21, 5,24,19, 3,11, 9, 8,18)( 2,14,15,26,22, 6,23,20, 4,12,10, 7, 17)$
$26$ $432$ $26$ $( 1,14,16,26,21, 6,24,20, 3,12, 9, 7,18, 2,13,15,25,22, 5,23,19, 4,11,10, 8,17 )$
$13^{2}$ $432$ $13$ $( 1, 3,25, 8,24,13,11,21,18,19,16, 9, 5)( 2, 4,26, 7,23,14,12,22,17,20,15,10, 6)$
$26$ $432$ $26$ $( 1, 4,25, 7,24,14,11,22,18,20,16,10, 5, 2, 3,26, 8,23,13,12,21,17,19,15, 9, 6 )$
$3^{8},1^{2}$ $624$ $3$ $( 1, 3,19)( 2, 4,20)( 5,18,13)( 6,17,14)( 9,24,21)(10,23,22)(11,16,25) (12,15,26)$
$6^{4},2$ $624$ $6$ $( 1, 4,19, 2, 3,20)( 5,17,13, 6,18,14)( 7, 8)( 9,23,21,10,24,22) (11,15,25,12,16,26)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $11232=2^{5} \cdot 3^{3} \cdot 13$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  11232.a
magma: IdentifyGroup(G);
 
Character table:

Size
2 P
3 P
13 P
Type

magma: CharacterTable(G);