Properties

Label 11T2
11T2 1 2 1->2 10 1->10 3 2->3 9 2->9 4 3->4 8 3->8 5 4->5 7 4->7 6 5->6 5->6 6->7 7->8 8->9 9->10 11 10->11 11->1
Degree $11$
Order $22$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive yes
$p$-group no
Group: $D_{11}$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(11, 2);
 
Copy content sage:G = TransitiveGroup(11, 2)
 
Copy content oscar:G = transitive_group(11, 2)
 
Copy content gap:G := TransitiveGroup(11, 2);
 

Group invariants

Abstract group:  $D_{11}$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $22=2 \cdot 11$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $11$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $2$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
CHM label:   $D(11)=11:2$
Parity:  $-1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
Primitive:  yes
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(11).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(11), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(11), G));
 
Generators:  $(1,2,3,4,5,6,7,8,9,10,11)$, $(1,10)(2,9)(3,8)(4,7)(5,6)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

22T2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{11}$ $1$ $1$ $0$ $()$
2A $2^{5},1$ $11$ $2$ $5$ $( 1, 9)( 2, 8)( 3, 7)( 4, 6)(10,11)$
11A1 $11$ $2$ $11$ $10$ $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
11A2 $11$ $2$ $11$ $10$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)$
11A3 $11$ $2$ $11$ $10$ $( 1, 9, 6, 3,11, 8, 5, 2,10, 7, 4)$
11A4 $11$ $2$ $11$ $10$ $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)$
11A5 $11$ $2$ $11$ $10$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)$

Malle's constant $a(G)$:     $1/5$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 11A1 11A2 11A3 11A4 11A5
Size 1 11 2 2 2 2 2
2 P 1A 1A 11A2 11A4 11A5 11A3 11A1
11 P 1A 2A 11A5 11A1 11A4 11A2 11A3
Type
22.1.1a R 1 1 1 1 1 1 1
22.1.1b R 1 −1 1 1 1 1 1
22.1.2a1 R 2 0 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−2+ζ112 ζ11−3+ζ113
22.1.2a2 R 2 0 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−5+ζ115 ζ11−2+ζ112
22.1.2a3 R 2 0 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−1+ζ11 ζ11−4+ζ114
22.1.2a4 R 2 0 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−3+ζ113 ζ11−1+ζ11
22.1.2a5 R 2 0 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−4+ζ114 ζ11−5+ζ115

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{11} + \left(-55 t^{10} - 55 t^{9} - 55 t^{8} - 55 t^{7} - 55 t^{6} - 55 t^{5} - 55 t^{4} - 55 t^{3} - 55 t^{2} - 55 t - 55\right) x^{9} + \left(330 t^{15} - 1320 t^{14} + 2222 t^{13} - 1320 t^{12} + 330 t^{11} - 330 t^{4} + 1320 t^{3} - 2222 t^{2} + 1320 t - 330\right) x^{8} + \left(-990 t^{20} + 12540 t^{19} - 15554 t^{18} + 13464 t^{17} + 18282 t^{16} - 4972 t^{15} - 154 t^{14} + 28864 t^{13} + 770 t^{12} + 14300 t^{11} + 13310 t^{10} + 14300 t^{9} + 770 t^{8} + 28864 t^{7} - 154 t^{6} - 4972 t^{5} + 18282 t^{4} + 13464 t^{3} - 15554 t^{2} + 12540 t - 990\right) x^{7} + \left(1848 t^{25} - 46200 t^{24} + 147840 t^{23} - 240944 t^{22} + 188760 t^{21} - 68728 t^{20} - 107448 t^{19} + 132616 t^{18} - 107448 t^{17} - 68728 t^{16} + 188760 t^{15} - 242792 t^{14} + 194040 t^{13} - 194040 t^{12} + 242792 t^{11} - 188760 t^{10} + 68728 t^{9} + 107448 t^{8} - 132616 t^{7} + 107448 t^{6} + 68728 t^{5} - 188760 t^{4} + 240944 t^{3} - 147840 t^{2} + 46200 t - 1848\right) x^{6} + \left(-2310 t^{30} + 94710 t^{29} - 677908 t^{28} + 1111396 t^{27} - 689018 t^{26} - 1935142 t^{25} + 2812216 t^{24} - 2485736 t^{23} - 1668238 t^{22} + 3252238 t^{21} - 4862748 t^{20} + 60038 t^{19} + 780516 t^{18} - 3744818 t^{17} - 786764 t^{16} - 232474 t^{15} - 786764 t^{14} - 3744818 t^{13} + 780516 t^{12} + 60038 t^{11} - 4862748 t^{10} + 3252238 t^{9} - 1668238 t^{8} - 2485736 t^{7} + 2812216 t^{6} - 1935142 t^{5} - 689018 t^{4} + 1111396 t^{3} - 677908 t^{2} + 94710 t - 2310\right) x^{5} + \left(1980 t^{35} - 120120 t^{34} + 1567720 t^{33} - 4215640 t^{32} + 8474356 t^{31} - 10888064 t^{30} - 2451460 t^{29} + 33005896 t^{28} - 19804796 t^{27} - 36185776 t^{26} + 56491996 t^{25} + 13734820 t^{24} - 57846932 t^{23} + 12169080 t^{22} + 60707636 t^{21} - 44660132 t^{20} - 8916732 t^{19} + 35457356 t^{18} - 35457356 t^{17} + 8916732 t^{16} + 44660132 t^{15} - 60707636 t^{14} - 12169080 t^{13} + 57846932 t^{12} - 13734820 t^{11} - 56491996 t^{10} + 36185776 t^{9} + 19804796 t^{8} - 33005896 t^{7} + 2451460 t^{6} + 10888064 t^{5} - 8474356 t^{4} + 4215640 t^{3} - 1567720 t^{2} + 120120 t - 1980\right) x^{4} + \left(-1155 t^{40} + 97020 t^{39} - 2024022 t^{38} + 10174780 t^{37} - 23000417 t^{36} - 9680792 t^{35} + 198948948 t^{34} - 344471688 t^{33} + 71065841 t^{32} + 561452100 t^{31} - 607596418 t^{30} - 137900840 t^{29} + 866082371 t^{28} - 344925944 t^{27} - 536571992 t^{26} + 646833484 t^{25} + 454066613 t^{24} - 967692484 t^{23} + 579809538 t^{22} + 633966432 t^{21} - 1025468345 t^{20} + 633966432 t^{19} + 579809538 t^{18} - 967692484 t^{17} + 454066613 t^{16} + 646833484 t^{15} - 536571992 t^{14} - 344925944 t^{13} + 866082371 t^{12} - 137900840 t^{11} - 607596418 t^{10} + 561452100 t^{9} + 71065841 t^{8} - 344471688 t^{7} + 198948948 t^{6} - 9680792 t^{5} - 23000417 t^{4} + 10174780 t^{3} - 2024022 t^{2} + 97020 t - 1155\right) x^{3} + \left(440 t^{45} - 48840 t^{44} + 1493624 t^{43} - 12974104 t^{42} + 33452144 t^{41} - 69153920 t^{40} + 374003256 t^{39} - 1131515528 t^{38} + 673971936 t^{37} + 2704214480 t^{36} - 5099877992 t^{35} - 835268632 t^{34} + 9423776032 t^{33} - 6295798432 t^{32} - 8375723576 t^{31} + 11916092584 t^{30} + 3012336624 t^{29} - 14632373184 t^{28} + 4074698232 t^{27} + 11275572792 t^{26} - 11092912160 t^{25} - 1194426816 t^{24} + 10258995384 t^{23} - 10258995384 t^{22} + 1194426816 t^{21} + 11092912160 t^{20} - 11275572792 t^{19} - 4074698232 t^{18} + 14632373184 t^{17} - 3012336624 t^{16} - 11916092584 t^{15} + 8375723576 t^{14} + 6295798432 t^{13} - 9423776032 t^{12} + 835268632 t^{11} + 5099877992 t^{10} - 2704214480 t^{9} - 673971936 t^{8} + 1131515528 t^{7} - 374003256 t^{6} + 69153920 t^{5} - 33452144 t^{4} + 12974104 t^{3} - 1493624 t^{2} + 48840 t - 440\right) x^{2} + \left(-99 t^{50} + 14025 t^{49} - 591965 t^{48} + 8091919 t^{47} - 29618050 t^{46} + 211247718 t^{45} - 1122306438 t^{44} + 1257893010 t^{43} + 5699281379 t^{42} - 17119562185 t^{41} + 7247576325 t^{40} + 31219489664 t^{39} - 49311571001 t^{38} - 1473014059 t^{37} + 71408342577 t^{36} - 62678308418 t^{35} - 32782834997 t^{34} + 88926274723 t^{33} - 29476203427 t^{32} - 76204989300 t^{31} + 85267322415 t^{30} + 18612786401 t^{29} - 121381405541 t^{28} + 79525196564 t^{27} + 61823584801 t^{26} - 145190816683 t^{25} + 61823584801 t^{24} + 79525196564 t^{23} - 121381405541 t^{22} + 18612786401 t^{21} + 85267322415 t^{20} - 76204989300 t^{19} - 29476203427 t^{18} + 88926274723 t^{17} - 32782834997 t^{16} - 62678308418 t^{15} + 71408342577 t^{14} - 1473014059 t^{13} - 49311571001 t^{12} + 31219489664 t^{11} + 7247576325 t^{10} - 17119562185 t^{9} + 5699281379 t^{8} + 1257893010 t^{7} - 1122306438 t^{6} + 211247718 t^{5} - 29618050 t^{4} + 8091919 t^{3} - 591965 t^{2} + 14025 t - 99\right) x + \left(10 t^{55} - 1760 t^{54} + 98098 t^{53} - 1959496 t^{52} + 11173668 t^{51} - 129339320 t^{50} + 835351814 t^{49} - 3075750832 t^{48} + 11541142888 t^{47} - 31365919200 t^{46} + 17893279690 t^{45} + 103588941142 t^{44} - 215068527828 t^{43} + 12717672462 t^{42} + 457763517574 t^{41} - 496485512468 t^{40} - 276414437640 t^{39} + 974143046338 t^{38} - 384746270590 t^{37} - 916639711856 t^{36} + 1072188005284 t^{35} + 334911204694 t^{34} - 1366462248414 t^{33} + 567873013972 t^{32} + 1028104315524 t^{31} - 1362860127046 t^{30} + 108652317378 t^{29} + 1251522175112 t^{28} - 1251522175112 t^{27} - 108652317378 t^{26} + 1362860127046 t^{25} - 1028104315524 t^{24} - 567873013972 t^{23} + 1366462248414 t^{22} - 334911204694 t^{21} - 1072188005284 t^{20} + 916639711856 t^{19} + 384746270590 t^{18} - 974143046338 t^{17} + 276414437640 t^{16} + 496485512468 t^{15} - 457763517574 t^{14} - 12717672462 t^{13} + 215068527828 t^{12} - 103588941142 t^{11} - 17893279690 t^{10} + 31365919200 t^{9} - 11541142888 t^{8} + 3075750832 t^{7} - 835351814 t^{6} + 129339320 t^{5} - 11173668 t^{4} + 1959496 t^{3} - 98098 t^{2} + 1760 t - 10\right)$ Copy content Toggle raw display