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Properties

Label 15T1
15T1 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 12 11->12 13 12->13 14 13->14 15 14->15 15->1
Degree $15$
Order $15$
Cyclic yes
Abelian yes
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{15}$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(15, 1);
 
Copy content sage:G = TransitiveGroup(15, 1)
 
Copy content oscar:G = transitive_group(15, 1)
 
Copy content gap:G := TransitiveGroup(15, 1);
 

Group invariants

Abstract group:  $C_{15}$
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:  $15=3 \cdot 5$
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 $n$:  $15$
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:   $C(15)=5[x]3$
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:  no
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);
 
$\card{\Aut(F/K)}$:  $15$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(15).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(15), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(15), G));
 
Generators:  $(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)$
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)
$3$:  $C_3$
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $C_5$

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 $1^{15}$ $1$ $1$ $0$ $()$
3A1 $3^{5}$ $1$ $3$ $10$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
3A-1 $3^{5}$ $1$ $3$ $10$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
5A1 $5^{3}$ $1$ $5$ $12$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
5A-1 $5^{3}$ $1$ $5$ $12$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
5A2 $5^{3}$ $1$ $5$ $12$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
5A-2 $5^{3}$ $1$ $5$ $12$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
15A1 $15$ $1$ $15$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$
15A-1 $15$ $1$ $15$ $14$ $( 1,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
15A2 $15$ $1$ $15$ $14$ $( 1, 3, 5, 7, 9,11,13,15, 2, 4, 6, 8,10,12,14)$
15A-2 $15$ $1$ $15$ $14$ $( 1,14,12,10, 8, 6, 4, 2,15,13,11, 9, 7, 5, 3)$
15A4 $15$ $1$ $15$ $14$ $( 1, 5, 9,13, 2, 6,10,14, 3, 7,11,15, 4, 8,12)$
15A-4 $15$ $1$ $15$ $14$ $( 1,12, 8, 4,15,11, 7, 3,14,10, 6, 2,13, 9, 5)$
15A7 $15$ $1$ $15$ $14$ $( 1, 8,15, 7,14, 6,13, 5,12, 4,11, 3,10, 2, 9)$
15A-7 $15$ $1$ $15$ $14$ $( 1, 9, 2,10, 3,11, 4,12, 5,13, 6,14, 7,15, 8)$

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 3A1 3A-1 5A1 5A-1 5A2 5A-2 15A1 15A-1 15A2 15A-2 15A4 15A-4 15A7 15A-7
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 P 1A 3A-1 3A1 5A2 5A-2 5A-1 5A1 15A2 15A-2 15A4 15A-4 15A-7 15A7 15A-1 15A1
5 P 1A 1A 1A 5A-2 5A2 5A1 5A-1 5A1 5A-1 5A2 5A-2 5A-1 5A1 5A2 5A-2
Type
15.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
15.1.1b1 C 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
15.1.1b2 C 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
15.1.1c1 C 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2
15.1.1c2 C 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52
15.1.1c3 C 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1
15.1.1c4 C 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5
15.1.1d1 C 1 ζ15−5 ζ155 ζ15−6 ζ156 ζ153 ζ15−3 ζ15−7 ζ157 ζ15 ζ15−1 ζ152 ζ15−2 ζ15−4 ζ154
15.1.1d2 C 1 ζ155 ζ15−5 ζ156 ζ15−6 ζ15−3 ζ153 ζ157 ζ15−7 ζ15−1 ζ15 ζ15−2 ζ152 ζ154 ζ15−4
15.1.1d3 C 1 ζ15−5 ζ155 ζ156 ζ15−6 ζ15−3 ζ153 ζ152 ζ15−2 ζ154 ζ15−4 ζ15−7 ζ157 ζ15−1 ζ15
15.1.1d4 C 1 ζ155 ζ15−5 ζ15−6 ζ156 ζ153 ζ15−3 ζ15−2 ζ152 ζ15−4 ζ154 ζ157 ζ15−7 ζ15 ζ15−1
15.1.1d5 C 1 ζ15−5 ζ155 ζ15−3 ζ153 ζ15−6 ζ156 ζ15−1 ζ15 ζ15−2 ζ152 ζ15−4 ζ154 ζ15−7 ζ157
15.1.1d6 C 1 ζ155 ζ15−5 ζ153 ζ15−3 ζ156 ζ15−6 ζ15 ζ15−1 ζ152 ζ15−2 ζ154 ζ15−4 ζ157 ζ15−7
15.1.1d7 C 1 ζ15−5 ζ155 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−4 ζ154 ζ157 ζ15−7 ζ15−1 ζ15 ζ152 ζ15−2
15.1.1d8 C 1 ζ155 ζ15−5 ζ15−3 ζ153 ζ15−6 ζ156 ζ154 ζ15−4 ζ15−7 ζ157 ζ15 ζ15−1 ζ15−2 ζ152

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^{15} + \left(-60 t^{8} - 60 t^{7} + 60 t^{5} + 60 t^{4} + 60 t^{3} - 60 t - 60\right) x^{13} + \left(-120 t^{12} - 180 t^{11} + 60 t^{10} + 300 t^{9} + 225 t^{8} + 45 t^{7} - 120 t^{6} - 285 t^{5} - 225 t^{4} + 75 t^{3} + 180 t^{2} + 45 t - 15\right) x^{12} + \left(630 t^{16} + 1260 t^{15} + 2880 t^{14} + 1215 t^{13} - 6120 t^{12} - 9495 t^{11} - 2430 t^{10} + 6120 t^{9} + 10035 t^{8} + 8370 t^{7} + 495 t^{6} - 8820 t^{5} - 8370 t^{4} - 1035 t^{3} + 2880 t^{2} + 2385 t + 1080\right) x^{11} + \left(1080 t^{20} + 2700 t^{19} + 12480 t^{18} + 12930 t^{17} - 32175 t^{16} - 66465 t^{15} - 8805 t^{14} + 69990 t^{13} + 73620 t^{12} + 30630 t^{11} - 26025 t^{10} - 84195 t^{9} - 70365 t^{8} + 19005 t^{7} + 62370 t^{6} + 27480 t^{5} - 5730 t^{4} - 10155 t^{3} - 6450 t^{2} - 1920 t + 135\right) x^{10} + \left(-3700 t^{24} - 11100 t^{23} - 14100 t^{22} + 275 t^{21} - 37575 t^{20} - 36975 t^{19} + 225075 t^{18} + 390075 t^{17} - 74925 t^{16} - 679850 t^{15} - 600600 t^{14} + 53325 t^{13} + 658075 t^{12} + 816825 t^{11} + 318150 t^{10} - 524975 t^{9} - 831300 t^{8} - 317175 t^{7} + 305325 t^{6} + 411150 t^{5} + 129300 t^{4} - 76600 t^{3} - 73725 t^{2} - 24975 t - 6700\right) x^{9} + \left(-3240 t^{28} - 11340 t^{27} - 98640 t^{26} - 203490 t^{25} - 197100 t^{24} + 300375 t^{23} + 1817415 t^{22} + 2249685 t^{21} - 1697175 t^{20} - 6546150 t^{19} - 4447485 t^{18} + 3812760 t^{17} + 9047745 t^{16} + 6332940 t^{15} - 1560600 t^{14} - 8621235 t^{13} - 8500590 t^{12} - 580455 t^{11} + 6773040 t^{10} + 6133500 t^{9} + 379620 t^{8} - 3002085 t^{7} - 2038500 t^{6} - 134955 t^{5} + 432000 t^{4} + 243270 t^{3} + 93510 t^{2} + 27180 t + 720\right) x^{8} + \left(13185 t^{32} + 52740 t^{31} + 51300 t^{30} - 87075 t^{29} - 578505 t^{28} - 656760 t^{27} + 2963880 t^{26} + 6132780 t^{25} - 4986495 t^{24} - 20683935 t^{23} - 6683280 t^{22} + 29587335 t^{21} + 34858335 t^{20} - 4146810 t^{19} - 41872125 t^{18} - 40252560 t^{17} - 1414695 t^{16} + 42565650 t^{15} + 48496590 t^{14} + 5455050 t^{13} - 40634880 t^{12} - 39277350 t^{11} + 482625 t^{10} + 27054840 t^{9} + 17056260 t^{8} - 3556410 t^{7} - 8796435 t^{6} - 3265260 t^{5} + 663105 t^{4} + 1012680 t^{3} + 381135 t^{2} + 65085 t + 12945\right) x^{7} + \left(200 t^{36} + 900 t^{35} + 317520 t^{34} + 1095510 t^{33} + 1342965 t^{32} - 1140795 t^{31} - 3152065 t^{30} - 3884910 t^{29} - 21420030 t^{28} - 31274945 t^{27} + 52225185 t^{26} + 168666195 t^{25} + 72309795 t^{24} - 240943365 t^{23} - 364346160 t^{22} - 48384720 t^{21} + 381366810 t^{20} + 457331610 t^{19} + 116200655 t^{18} - 339167805 t^{17} - 516940785 t^{16} - 215609345 t^{15} + 286864620 t^{14} + 441730035 t^{13} + 134916615 t^{12} - 186544620 t^{11} - 199421190 t^{10} - 44474735 t^{9} + 50414070 t^{8} + 50004780 t^{7} + 16419885 t^{6} - 5054205 t^{5} - 6743880 t^{4} - 2287290 t^{3} - 354210 t^{2} - 62295 t - 2165\right) x^{6} + \left(-24408 t^{40} - 122040 t^{39} - 205020 t^{38} + 72045 t^{37} + 3385035 t^{36} + 8331327 t^{35} + 2589660 t^{34} - 27944505 t^{33} - 90865395 t^{32} - 82737360 t^{31} + 223808670 t^{30} + 604269090 t^{29} + 157368825 t^{28} - 1163241225 t^{27} - 1495468575 t^{26} + 394109793 t^{25} + 2419552845 t^{24} + 1776120525 t^{23} - 870588225 t^{22} - 2592544050 t^{21} - 1962964782 t^{20} + 367699005 t^{19} + 2507461785 t^{18} + 2199878190 t^{17} - 430605405 t^{16} - 2171789397 t^{15} - 1167378615 t^{14} + 592280370 t^{13} + 923587155 t^{12} + 258402465 t^{11} - 182301030 t^{10} - 186755535 t^{9} - 64639125 t^{8} + 14536350 t^{7} + 28278000 t^{6} + 11192247 t^{5} - 485235 t^{4} - 1707165 t^{3} - 506160 t^{2} - 50130 t - 8298\right) x^{5} + \left(16800 t^{44} + 92400 t^{43} - 452700 t^{42} - 2843250 t^{41} - 4112775 t^{40} + 4765050 t^{39} + 27755700 t^{38} + 28170600 t^{37} - 102646500 t^{36} - 236976075 t^{35} + 262363950 t^{34} + 1125675675 t^{33} + 57420600 t^{32} - 3088013325 t^{31} - 2704498050 t^{30} + 4001374725 t^{29} + 7992122550 t^{28} + 602625375 t^{27} - 9946040775 t^{26} - 9441360675 t^{25} + 1666742775 t^{24} + 11453091450 t^{23} + 10965380850 t^{22} - 429567450 t^{21} - 12864541575 t^{20} - 12042379200 t^{19} + 1917103500 t^{18} + 11905446750 t^{17} + 7498017150 t^{16} - 2721475875 t^{15} - 6587871300 t^{14} - 3024488325 t^{13} + 1436258100 t^{12} + 2380219425 t^{11} + 808231575 t^{10} - 429699300 t^{9} - 467552025 t^{8} - 124469775 t^{7} + 30568350 t^{6} + 36361275 t^{5} + 15433875 t^{4} + 3409350 t^{3} + 299925 t^{2} + 41175 t + 1650\right) x^{4} + \left(13200 t^{48} + 79200 t^{47} + 437340 t^{46} + 1327080 t^{45} - 5012835 t^{44} - 27799605 t^{43} - 26830255 t^{42} + 80481690 t^{41} + 168540270 t^{40} - 12450260 t^{39} + 109013520 t^{38} + 737340720 t^{37} - 1339533575 t^{36} - 6347785410 t^{35} - 1941791400 t^{34} + 17001022975 t^{33} + 21245368350 t^{32} - 13775116815 t^{31} - 47800578205 t^{30} - 23567551440 t^{29} + 37715855355 t^{28} + 63315366830 t^{27} + 26209217235 t^{26} - 39249162270 t^{25} - 75734133265 t^{24} - 40039329345 t^{23} + 41861792655 t^{22} + 79317923110 t^{21} + 30950287005 t^{20} - 37936004040 t^{19} - 52129249820 t^{18} - 16237533795 t^{17} + 18986848725 t^{16} + 24615561385 t^{15} + 8602644600 t^{14} - 6714010500 t^{13} - 8938103500 t^{12} - 2807864610 t^{11} + 1753988130 t^{10} + 1876493165 t^{9} + 483525720 t^{8} - 184403730 t^{7} - 169302535 t^{6} - 47821380 t^{5} - 3232560 t^{4} + 1231195 t^{3} + 235845 t^{2} + 5850 t + 1090\right) x^{3} + \left(-23040 t^{52} - 149760 t^{51} + 159840 t^{50} + 2768400 t^{49} + 6505380 t^{48} - 1174770 t^{47} - 59365665 t^{46} - 133294860 t^{45} + 26675730 t^{44} + 607564755 t^{43} + 1447157115 t^{42} + 847023660 t^{41} - 5568529770 t^{40} - 13667186970 t^{39} + 839278800 t^{38} + 41822203410 t^{37} + 42910619625 t^{36} - 45310419360 t^{35} - 120401226000 t^{34} - 40846091130 t^{33} + 127328370795 t^{32} + 171217434015 t^{31} + 26164735560 t^{30} - 159380427780 t^{29} - 206043574455 t^{28} - 54349809300 t^{27} + 170135503200 t^{26} + 232116252930 t^{25} + 52766557695 t^{24} - 161095214430 t^{23} - 178553208690 t^{22} - 27502194285 t^{21} + 97263081495 t^{20} + 94818068145 t^{19} + 20467274625 t^{18} - 35692944435 t^{17} - 41063118975 t^{16} - 15369399990 t^{15} + 9131103675 t^{14} + 14886904680 t^{13} + 6315046605 t^{12} - 2142148140 t^{11} - 3507943185 t^{10} - 1194071895 t^{9} + 298009755 t^{8} + 372708090 t^{7} + 107273385 t^{6} - 4160295 t^{5} - 10277145 t^{4} - 2241225 t^{3} - 86355 t^{2} + 540 t - 45\right) x^{2} + \left(11520 t^{56} + 80640 t^{55} - 301200 t^{54} - 2990400 t^{53} - 154020 t^{52} + 35171160 t^{51} + 56967450 t^{50} - 132590400 t^{49} - 536280630 t^{48} - 181895655 t^{47} + 2532227925 t^{46} + 4454562180 t^{45} - 5536603170 t^{44} - 22817127675 t^{43} - 4699099605 t^{42} + 59221551870 t^{41} + 63520348800 t^{40} - 69106855545 t^{39} - 172833737895 t^{38} - 34715566275 t^{37} + 211247293395 t^{36} + 228382711695 t^{35} - 26271122550 t^{34} - 274213984245 t^{33} - 272910443445 t^{32} + 3211528830 t^{31} + 326607810030 t^{30} + 334140757920 t^{29} - 30382903200 t^{28} - 341304237750 t^{27} - 251848711980 t^{26} + 55225070925 t^{25} + 214901473020 t^{24} + 142405043625 t^{23} - 13360037370 t^{22} - 102346978320 t^{21} - 75492598275 t^{20} + 2469435195 t^{19} + 37338656070 t^{18} + 20079709875 t^{17} + 1878723930 t^{16} - 751083720 t^{15} - 3304563975 t^{14} - 6213636930 t^{13} - 2666079405 t^{12} + 2109230550 t^{11} + 2240696805 t^{10} + 242608680 t^{9} - 495253725 t^{8} - 215246355 t^{7} + 8760315 t^{6} + 24669690 t^{5} + 4948530 t^{4} + 60780 t^{3} - 27345 t^{2} - 345 t - 30\right) x + \left(-2048 t^{60} - 15360 t^{59} + 90240 t^{58} + 846400 t^{57} - 1102560 t^{56} - 17683152 t^{55} - 2959840 t^{54} + 180214080 t^{53} + 154431780 t^{52} - 1022883600 t^{51} - 1263236466 t^{50} + 3942940950 t^{49} + 6213845950 t^{48} - 11173670175 t^{47} - 23176445145 t^{46} + 20764345420 t^{45} + 65935661565 t^{44} - 11858463945 t^{43} - 131379609490 t^{42} - 53783502285 t^{41} + 157890917070 t^{40} + 165640558490 t^{39} - 55430523570 t^{38} - 197450142765 t^{37} - 143867849390 t^{36} + 21611493018 t^{35} + 230457879360 t^{34} + 260242218310 t^{33} - 64372979775 t^{32} - 365154227745 t^{31} - 187296989843 t^{30} + 193507553775 t^{29} + 250218164940 t^{28} + 39783925760 t^{27} - 103486054815 t^{26} - 117246285354 t^{25} - 53493543790 t^{24} + 60674491470 t^{23} + 99306585240 t^{22} + 1928590160 t^{21} - 73305101715 t^{20} - 22191493740 t^{19} + 40467241645 t^{18} + 20630725440 t^{17} - 18553097160 t^{16} - 14412725890 t^{15} + 5531398425 t^{14} + 7418059200 t^{13} - 138446950 t^{12} - 2381262570 t^{11} - 583621998 t^{10} + 376213180 t^{9} + 196810920 t^{8} - 5886615 t^{7} - 21633445 t^{6} - 4016652 t^{5} + 167550 t^{4} + 84360 t^{3} + 3315 t^{2} - 165 t - 1\right)$ Copy content Toggle raw display