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Properties

Label 26T47
26T47 1 3 1->3 14 1->14 2 4 2->4 13 2->13 24 3->24 26 3->26 23 4->23 25 4->25 5 5->1 18 5->18 6 6->2 17 6->17 7 16 7->16 22 7->22 8 15 8->15 21 8->21 9 10 9->10 9->17 10->18 11 11->8 12 12->7 13->5 13->22 14->6 14->21 15->14 15->26 16->13 16->25 17->12 20 17->20 18->11 19 18->19 19->2 19->23 20->1 20->24 21->6 21->19 22->5 22->20 23->9 23->15 24->10 24->16 25->4 25->8 26->3 26->7
Degree $26$
Order $11232$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\GL(3,3)$

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Copy content magma:G := TransitiveGroup(26, 47);
 

Group invariants

Abstract group:  $\GL(3,3)$
Copy content magma:IdentifyGroup(G);
 
Order:  $11232=2^{5} \cdot 3^{3} \cdot 13$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $26$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $47$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,14,6,17,12,7,22,20)(2,13,5,18,11,8,21,19)(3,24,16,25,4,23,15,26)(9,10)$, $(1,3,26,7,16,13,22,5)(2,4,25,8,15,14,21,6)(9,17,20,24,10,18,19,23)$
Copy content magma:Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

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

Low degree siblings

26T47, 26T48 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{26}$ $1$ $1$ $0$ $()$
2A $2^{13}$ $1$ $2$ $13$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)$
2B $2^{12},1^{2}$ $117$ $2$ $12$ $( 1,16)( 2,15)( 3, 4)( 5, 6)( 7,24)( 8,23)( 9,10)(13,18)(14,17)(19,25)(20,26)(21,22)$
2C $2^{9},1^{8}$ $117$ $2$ $9$ $( 3, 5)( 4, 6)( 7,26)( 8,25)( 9,10)(11,16)(12,15)(17,19)(18,20)$
3A $3^{6},1^{8}$ $104$ $3$ $12$ $( 5,25,18)( 6,26,17)( 9,15,11)(10,16,12)(13,21,23)(14,22,24)$
3B $3^{8},1^{2}$ $624$ $3$ $16$ $( 3,20, 8)( 4,19, 7)( 5,13,10)( 6,14, 9)(11,17,24)(12,18,23)(15,26,22)(16,25,21)$
4A $4^{6},1^{2}$ $702$ $4$ $18$ $( 1,17,16,14)( 2,18,15,13)( 3,10, 4, 9)( 5,21, 6,22)( 7,20,24,26)( 8,19,23,25)$
4B $4^{6},2$ $702$ $4$ $19$ $( 1, 2)( 3,25, 7, 6)( 4,26, 8, 5)( 9,23,10,24)(11,21,16,14)(12,22,15,13)(17,20,18,19)$
6A $6^{3},2^{4}$ $104$ $6$ $19$ $( 1, 2)( 3, 4)( 5,17,25, 6,18,26)( 7, 8)( 9,12,15,10,11,16)(13,24,21,14,23,22)(19,20)$
6B $6^{4},2$ $624$ $6$ $21$ $( 1, 2)( 3, 7,20, 4, 8,19)( 5, 9,13, 6,10,14)(11,23,17,12,24,18)(15,21,26,16,22,25)$
6C $6^{2},3^{2},2^{3},1^{2}$ $936$ $6$ $17$ $( 3,18, 8, 5,20,25)( 4,17, 7, 6,19,26)( 9,10)(11,16)(12,15)(13,21,23)(14,22,24)$
6D $6^{3},2^{3},1^{2}$ $936$ $6$ $18$ $( 1, 2)( 5,17,25, 6,18,26)( 7,19)( 8,20)( 9,13,15,21,11,23)(10,14,16,22,12,24)$
8A1 $8^{3},2$ $702$ $8$ $22$ $( 1,24,17,26,16, 7,14,20)( 2,23,18,25,15, 8,13,19)( 3,22,10, 5, 4,21, 9, 6)(11,12)$
8A-1 $8^{3},2$ $702$ $8$ $22$ $( 1,20,14, 7,16,26,17,24)( 2,19,13, 8,15,25,18,23)( 3, 6, 9,21, 4, 5,10,22)(11,12)$
8B1 $8^{3},1^{2}$ $702$ $8$ $21$ $( 1, 6,17,25, 2, 5,18,26)( 3, 9,23,15,11, 8,21,20)( 4,10,24,16,12, 7,22,19)$
8B-1 $8^{3},1^{2}$ $702$ $8$ $21$ $( 1,26,18, 5, 2,25,17, 6)( 3,20,21, 8,11,15,23, 9)( 4,19,22, 7,12,16,24,10)$
13A1 $13^{2}$ $432$ $13$ $24$ $( 1,11, 8, 4,14,10, 6,19,18,25,22,15,24)( 2,12, 7, 3,13, 9, 5,20,17,26,21,16,23)$
13A-1 $13^{2}$ $432$ $13$ $24$ $( 1,24,15,22,25,18,19, 6,10,14, 4, 8,11)( 2,23,16,21,26,17,20, 5, 9,13, 3, 7,12)$
13A2 $13^{2}$ $432$ $13$ $24$ $( 1, 8,14, 6,18,22,24,11, 4,10,19,25,15)( 2, 7,13, 5,17,21,23,12, 3, 9,20,26,16)$
13A-2 $13^{2}$ $432$ $13$ $24$ $( 1,15,25,19,10, 4,11,24,22,18, 6,14, 8)( 2,16,26,20, 9, 3,12,23,21,17, 5,13, 7)$
26A1 $26$ $432$ $26$ $25$ $( 1,20,11,17, 8,26, 4,21,14,16,10,23, 6, 2,19,12,18, 7,25, 3,22,13,15, 9,24, 5)$
26A-1 $26$ $432$ $26$ $25$ $( 1, 5,24, 9,15,13,22, 3,25, 7,18,12,19, 2, 6,23,10,16,14,21, 4,26, 8,17,11,20)$
26A5 $26$ $432$ $26$ $25$ $( 1,26,10,12,22, 5, 8,16,19, 3,24,17,14, 2,25, 9,11,21, 6, 7,15,20, 4,23,18,13)$
26A-5 $26$ $432$ $26$ $25$ $( 1,13,18,23, 4,20,15, 7, 6,21,11, 9,25, 2,14,17,24, 3,19,16, 8, 5,22,12,10,26)$

Malle's constant $a(G)$:     $1/9$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 4A 4B 6A 6B 6C 6D 8A1 8A-1 8B1 8B-1 13A1 13A-1 13A2 13A-2 26A1 26A-1 26A5 26A-5
Size 1 1 117 117 104 624 702 702 104 624 936 936 702 702 702 702 432 432 432 432 432 432 432 432
2 P 1A 1A 1A 1A 3A 3B 2B 2B 3A 3B 3A 3A 4A 4A 4A 4A 13A2 13A-2 13A-1 13A1 13A1 13A-1 13A2 13A-2
3 P 1A 2A 2B 2C 1A 1A 4A 4B 2A 2A 2C 2B 8A1 8A-1 8B1 8B-1 13A1 13A-1 13A2 13A-2 26A1 26A-1 26A5 26A-5
13 P 1A 2A 2B 2C 3A 3B 4A 4B 6A 6B 6C 6D 8A-1 8A1 8B-1 8B1 1A 1A 1A 1A 2A 2A 2A 2A
Type
11232.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
11232.a.1b R 1 −1 1 −1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
11232.a.12a R 12 12 4 4 3 0 0 0 3 0 1 1 0 0 0 0 −1 −1 −1 −1 −1 −1 −1 −1
11232.a.12b R 12 −12 4 −4 3 0 0 0 −3 0 −1 1 0 0 0 0 −1 −1 −1 −1 1 1 1 1
11232.a.13a R 13 13 −3 −3 4 1 1 1 4 1 0 0 −1 −1 −1 −1 0 0 0 0 0 0 0 0
11232.a.13b R 13 −13 −3 3 4 1 1 −1 −4 −1 0 0 −1 −1 1 1 0 0 0 0 0 0 0 0
11232.a.16a1 C 16 16 0 0 −2 1 0 0 −2 1 0 0 0 0 0 0 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134
11232.a.16a2 C 16 16 0 0 −2 1 0 0 −2 1 0 0 0 0 0 0 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133
11232.a.16a3 C 16 16 0 0 −2 1 0 0 −2 1 0 0 0 0 0 0 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136
11232.a.16a4 C 16 16 0 0 −2 1 0 0 −2 1 0 0 0 0 0 0 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2
11232.a.16b1 C 16 −16 0 0 −2 1 0 0 2 −1 0 0 0 0 0 0 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 −ζ13−6−ζ13−5−ζ13−2 −ζ132−ζ135−ζ136 −ζ13−4−ζ13−ζ133 −ζ13−3−ζ13−1−ζ134
11232.a.16b2 C 16 −16 0 0 −2 1 0 0 2 −1 0 0 0 0 0 0 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 −ζ132−ζ135−ζ136 −ζ13−6−ζ13−5−ζ13−2 −ζ13−3−ζ13−1−ζ134 −ζ13−4−ζ13−ζ133
11232.a.16b3 C 16 −16 0 0 −2 1 0 0 2 −1 0 0 0 0 0 0 ζ13−3+ζ13−1+ζ134 ζ13−4+ζ13+ζ133 ζ132+ζ135+ζ136 ζ13−6+ζ13−5+ζ13−2 −ζ13−3−ζ13−1−ζ134 −ζ13−4−ζ13−ζ133 −ζ13−6−ζ13−5−ζ13−2 −ζ132−ζ135−ζ136
11232.a.16b4 C 16 −16 0 0 −2 1 0 0 2 −1 0 0 0 0 0 0 ζ13−4+ζ13+ζ133 ζ13−3+ζ13−1+ζ134 ζ13−6+ζ13−5+ζ13−2 ζ132+ζ135+ζ136 −ζ13−4−ζ13−ζ133 −ζ13−3−ζ13−1−ζ134 −ζ132−ζ135−ζ136 −ζ13−6−ζ13−5−ζ13−2
11232.a.26a R 26 26 2 2 −1 −1 2 2 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
11232.a.26b R 26 −26 2 −2 −1 −1 2 −2 1 1 1 −1 0 0 0 0 0 0 0 0 0 0 0 0
11232.a.26c1 C 26 −26 −2 2 −1 −1 0 0 1 1 −1 1 −ζ8−ζ83 ζ8+ζ83 ζ8+ζ83 −ζ8−ζ83 0 0 0 0 0 0 0 0
11232.a.26c2 C 26 −26 −2 2 −1 −1 0 0 1 1 −1 1 ζ8+ζ83 −ζ8−ζ83 −ζ8−ζ83 ζ8+ζ83 0 0 0 0 0 0 0 0
11232.a.26d1 C 26 26 −2 −2 −1 −1 0 0 −1 −1 1 1 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 0 0 0 0 0 0 0 0
11232.a.26d2 C 26 26 −2 −2 −1 −1 0 0 −1 −1 1 1 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 0 0 0 0 0 0 0 0
11232.a.27a R 27 27 3 3 0 0 −1 −1 0 0 0 0 −1 −1 −1 −1 1 1 1 1 1 1 1 1
11232.a.27b R 27 −27 3 −3 0 0 −1 1 0 0 0 0 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
11232.a.39a R 39 −39 −1 1 3 0 −1 1 −3 0 1 −1 1 1 −1 −1 0 0 0 0 0 0 0 0
11232.a.39b R 39 39 −1 −1 3 0 −1 −1 3 0 −1 −1 1 1 1 1 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed