Properties

Label 11T1
11T1 1 2 1->2 3 2->3 4 3->4 5 4->5 6 5->6 7 6->7 8 7->8 9 8->9 10 9->10 11 10->11 11->1
Degree $11$
Order $11$
Cyclic yes
Abelian yes
Solvable yes
Transitivity $1$
Primitive yes
$p$-group yes
Group: $C_{11}$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(11, 1);
 
Copy content sage:G = TransitiveGroup(11, 1)
 
Copy content oscar:G = transitive_group(11, 1)
 
Copy content gap:G := TransitiveGroup(11, 1);
 

Group invariants

Abstract group:  $C_{11}$
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:  $11$ (is prime)
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:  yes
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:  yes
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:  $1$
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 $t$:  $1$
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:   $C(11)=11$
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);
 
$\card{\Aut(F/K)}$:  $11$
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:  $(1,2,3,4,5,6,7,8,9,10,11)$
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

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

Malle's constant $a(G)$:     $1/10$

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 11A1 11A-1 11A2 11A-2 11A3 11A-3 11A4 11A-4 11A5 11A-5
Size 1 1 1 1 1 1 1 1 1 1 1
11 P 1A 11A2 11A-2 11A4 11A-4 11A-5 11A5 11A-3 11A3 11A-1 11A1
Type
11.1.1a R 1 1 1 1 1 1 1 1 1 1 1
11.1.1b1 C 1 ζ11−5 ζ115 ζ11 ζ11−1 ζ11−4 ζ114 ζ112 ζ11−2 ζ11−3 ζ113
11.1.1b2 C 1 ζ115 ζ11−5 ζ11−1 ζ11 ζ114 ζ11−4 ζ11−2 ζ112 ζ113 ζ11−3
11.1.1b3 C 1 ζ11−4 ζ114 ζ113 ζ11−3 ζ11−1 ζ11 ζ11−5 ζ115 ζ112 ζ11−2
11.1.1b4 C 1 ζ114 ζ11−4 ζ11−3 ζ113 ζ11 ζ11−1 ζ115 ζ11−5 ζ11−2 ζ112
11.1.1b5 C 1 ζ11−3 ζ113 ζ115 ζ11−5 ζ112 ζ11−2 ζ11−1 ζ11 ζ11−4 ζ114
11.1.1b6 C 1 ζ113 ζ11−3 ζ11−5 ζ115 ζ11−2 ζ112 ζ11 ζ11−1 ζ114 ζ11−4
11.1.1b7 C 1 ζ11−2 ζ112 ζ11−4 ζ114 ζ115 ζ11−5 ζ113 ζ11−3 ζ11 ζ11−1
11.1.1b8 C 1 ζ112 ζ11−2 ζ114 ζ11−4 ζ11−5 ζ115 ζ11−3 ζ113 ζ11−1 ζ11
11.1.1b9 C 1 ζ11−1 ζ11 ζ11−2 ζ112 ζ11−3 ζ113 ζ11−4 ζ114 ζ11−5 ζ115
11.1.1b10 C 1 ζ11 ζ11−1 ζ112 ζ11−2 ζ113 ζ11−3 ζ114 ζ11−4 ζ115 ζ11−5

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} + 495 t^{14} - 165 t^{13} - 110 t^{12} + 154 t^{11} - 121 t^{10} + 121 t^{9} - 121 t^{8} + 121 t^{7} - 121 t^{6} + 121 t^{5} - 451 t^{4} + 616 t^{3} - 286 t^{2} + 11 t + 33\right) x^{8} + \left(-990 t^{20} + 1980 t^{19} + 660 t^{18} - 4510 t^{17} + 7029 t^{16} - 8096 t^{15} + 7711 t^{14} - 7568 t^{13} + 8030 t^{12} - 8492 t^{11} + 9317 t^{10} - 10307 t^{9} + 11297 t^{8} - 8657 t^{7} + 4807 t^{6} - 2288 t^{5} + 1221 t^{4} - 1606 t^{3} + 1749 t^{2} - 1287 t + 825\right) x^{7} + \left(-1848 t^{25} + 4620 t^{24} + 4620 t^{23} - 23870 t^{22} + 37268 t^{21} - 39930 t^{20} + 30250 t^{19} - 16335 t^{18} + 6655 t^{17} + 2541 t^{16} - 8833 t^{15} + 11792 t^{14} - 13750 t^{13} + 25454 t^{12} - 45562 t^{11} + 58564 t^{10} - 61226 t^{9} + 51546 t^{8} - 37631 t^{7} + 27951 t^{6} - 18755 t^{5} + 12463 t^{4} - 7656 t^{3} + 2926 t^{2} - 462 t - 396\right) x^{6} + \left(-2310 t^{30} + 6930 t^{29} + 11550 t^{28} - 61600 t^{27} + 91553 t^{26} - 68255 t^{25} - 19426 t^{24} + 151162 t^{23} - 253264 t^{22} + 325369 t^{21} - 397122 t^{20} + 454938 t^{19} - 517451 t^{18} + 580910 t^{17} - 626318 t^{16} + 632863 t^{15} - 588808 t^{14} + 493185 t^{13} - 365354 t^{12} + 268840 t^{11} - 201707 t^{10} + 129954 t^{9} - 69828 t^{8} + 2695 t^{7} + 42284 t^{6} - 37642 t^{5} + 14234 t^{4} + 6523 t^{3} - 14465 t^{2} + 10560 t - 4972\right) x^{5} + \left(-1980 t^{35} + 6930 t^{34} + 16170 t^{33} - 93170 t^{32} + 127050 t^{31} - 10285 t^{30} - 287375 t^{29} + 736890 t^{28} - 1098317 t^{27} + 1209516 t^{26} - 1183138 t^{25} + 1069750 t^{24} - 987866 t^{23} + 1001264 t^{22} - 919721 t^{21} + 589512 t^{20} - 5929 t^{19} - 697202 t^{18} + 1346004 t^{17} - 1713481 t^{16} + 1752201 t^{15} - 1696783 t^{14} + 1597068 t^{13} - 1530672 t^{12} + 1528098 t^{11} - 1368268 t^{10} + 1004179 t^{9} - 537361 t^{8} + 131890 t^{7} + 67397 t^{6} - 73447 t^{5} + 968 t^{4} + 28072 t^{3} - 16379 t^{2} + 5841 t + 1287\right) x^{4} + \left(-1155 t^{40} + 4620 t^{39} + 13860 t^{38} - 87010 t^{37} + 105259 t^{36} + 114774 t^{35} - 608729 t^{34} + 1317624 t^{33} - 1802900 t^{32} + 1425017 t^{31} - 301653 t^{30} - 1189991 t^{29} + 2719299 t^{28} - 3719705 t^{27} + 4516292 t^{26} - 5809122 t^{25} + 7650544 t^{24} - 9979772 t^{23} + 12188616 t^{22} - 13094499 t^{21} + 12228260 t^{20} - 10281007 t^{19} + 8215581 t^{18} - 6667397 t^{17} + 5891567 t^{16} - 5154391 t^{15} + 3809047 t^{14} - 2110647 t^{13} + 258401 t^{12} + 1220692 t^{11} - 1628539 t^{10} + 1140183 t^{9} - 316294 t^{8} - 258643 t^{7} + 280984 t^{6} - 37928 t^{5} - 94633 t^{4} + 60368 t^{3} + 16643 t^{2} - 33616 t + 12760\right) x^{3} + \left(-440 t^{45} + 1980 t^{44} + 7260 t^{43} - 49610 t^{42} + 51304 t^{41} + 157784 t^{40} - 593384 t^{39} + 1134617 t^{38} - 1294337 t^{37} - 36058 t^{36} + 3119985 t^{35} - 6958897 t^{34} + 10420091 t^{33} - 12011791 t^{32} + 11646492 t^{31} - 11272723 t^{30} + 11780197 t^{29} - 13051060 t^{28} + 14528349 t^{27} - 14053182 t^{26} + 9678669 t^{25} - 1973510 t^{24} - 6693577 t^{23} + 13410331 t^{22} - 15568223 t^{21} + 14174061 t^{20} - 13036298 t^{19} + 13511828 t^{18} - 15098138 t^{17} + 16633991 t^{16} - 15965829 t^{15} + 12642806 t^{14} - 7933728 t^{13} + 3139400 t^{12} + 92378 t^{11} - 650254 t^{10} - 420959 t^{9} + 1186647 t^{8} - 1009503 t^{7} + 258456 t^{6} + 341341 t^{5} - 308066 t^{4} + 29161 t^{3} + 58685 t^{2} - 24398 t - 924\right) x^{2} + \left(-99 t^{50} + 495 t^{49} + 2145 t^{48} - 15895 t^{47} + 13398 t^{46} + 87010 t^{45} - 283965 t^{44} + 461615 t^{43} - 321739 t^{42} - 1031041 t^{41} + 4221569 t^{40} - 8377710 t^{39} + 11805167 t^{38} - 11790614 t^{37} + 6391044 t^{36} + 1794540 t^{35} - 8986263 t^{34} + 13561163 t^{33} - 15280606 t^{32} + 17728678 t^{31} - 25256935 t^{30} + 37368067 t^{29} - 51532921 t^{28} + 63516596 t^{27} - 66692494 t^{26} + 59130621 t^{25} - 46209130 t^{24} + 35000284 t^{23} - 30102974 t^{22} + 30980961 t^{21} - 31204206 t^{20} + 25549997 t^{19} - 15599078 t^{18} + 4598209 t^{17} + 4227542 t^{16} - 7201084 t^{15} + 4809695 t^{14} - 122100 t^{13} - 3730144 t^{12} + 3873056 t^{11} - 1108965 t^{10} - 1422476 t^{9} + 1943744 t^{8} - 913429 t^{7} - 78628 t^{6} + 347765 t^{5} - 133496 t^{4} - 81840 t^{3} + 36025 t^{2} + 28446 t - 10989\right) x + \left(-10 t^{55} + 55 t^{54} + 275 t^{53} - 2200 t^{52} + 1408 t^{51} + 18040 t^{50} - 54076 t^{49} + 67837 t^{48} + 15950 t^{47} - 464970 t^{46} + 1603921 t^{45} - 3173264 t^{44} + 4443461 t^{43} - 4311219 t^{42} + 859617 t^{41} + 6240883 t^{40} - 14243592 t^{39} + 20567140 t^{38} - 23116258 t^{37} + 21961170 t^{36} - 22676720 t^{35} + 28608052 t^{34} - 37936339 t^{33} + 48499066 t^{32} - 54708742 t^{31} + 51366546 t^{30} - 40655054 t^{29} + 27633441 t^{28} - 17720989 t^{27} + 16652724 t^{26} - 22791571 t^{25} + 26285875 t^{24} - 22113685 t^{23} + 11522796 t^{22} + 2234727 t^{21} - 11272107 t^{20} + 12050544 t^{19} - 8476952 t^{18} + 3152215 t^{17} + 588588 t^{16} + 892441 t^{15} - 5621627 t^{14} + 9300555 t^{13} - 9963118 t^{12} + 7150731 t^{11} - 2685540 t^{10} - 10252 t^{9} + 681208 t^{8} - 717816 t^{7} + 392535 t^{6} - 204600 t^{5} + 191015 t^{4} - 20229 t^{3} - 61677 t^{2} + 19305 t + 243\right)$ Copy content Toggle raw display