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Properties

Label 26T48
26T48 1 2 1->2 15 1->15 16 2->16 3 7 3->7 23 3->23 4 8 4->8 24 4->24 5 6 5->6 20 5->20 19 6->19 13 7->13 21 7->21 14 8->14 22 8->22 9 9->4 12 9->12 10 10->3 11 10->11 11->15 26 11->26 12->16 25 12->25 13->4 13->10 14->3 14->9 15->9 15->24 16->10 16->23 17 17->13 17->25 18 18->14 18->26 19->17 19->22 20->18 20->21 21->2 21->7 22->1 22->8 23->11 23->19 24->12 24->20 25->6 26->5
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, 48);
 

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 $t$:  $48$
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:  $(1,2)(3,23,19,22,8,14)(4,24,20,21,7,13)(5,6)(9,12,16,10,11,15)(17,25)(18,26)$, $(1,15,24,12,25,6,19,17,13,10,3,7,21,2,16,23,11,26,5,20,18,14,9,4,8,22)$
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: $C_2$

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

Malle's constant $a(G)$:     $1/8$

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