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Properties

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

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Copy content magma:G := TransitiveGroup(14, 1);
 

Group invariants

Abstract group:  $C_{14}$
Copy content magma:IdentifyGroup(G);
 
Order:  $14=2 \cdot 7$
Copy content magma:Order(G);
 
Cyclic:  yes
Copy content magma:IsCyclic(G);
 
Abelian:  yes
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $1$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $14$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $C(14)=7[x]2$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $14$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2,3,4,5,6,7,8,9,10,11,12,13,14)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$7$:  $C_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 7: $C_7$

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 $1^{14}$ $1$ $1$ $0$ $()$
2A $2^{7}$ $1$ $2$ $7$ $( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
7A1 $7^{2}$ $1$ $7$ $12$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$
7A-1 $7^{2}$ $1$ $7$ $12$ $( 1,13,11, 9, 7, 5, 3)( 2,14,12,10, 8, 6, 4)$
7A2 $7^{2}$ $1$ $7$ $12$ $( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$
7A-2 $7^{2}$ $1$ $7$ $12$ $( 1,11, 7, 3,13, 9, 5)( 2,12, 8, 4,14,10, 6)$
7A3 $7^{2}$ $1$ $7$ $12$ $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$
7A-3 $7^{2}$ $1$ $7$ $12$ $( 1, 9, 3,11, 5,13, 7)( 2,10, 4,12, 6,14, 8)$
14A1 $14$ $1$ $14$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14)$
14A-1 $14$ $1$ $14$ $13$ $( 1,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
14A3 $14$ $1$ $14$ $13$ $( 1, 4, 7,10,13, 2, 5, 8,11,14, 3, 6, 9,12)$
14A-3 $14$ $1$ $14$ $13$ $( 1,12, 9, 6, 3,14,11, 8, 5, 2,13,10, 7, 4)$
14A5 $14$ $1$ $14$ $13$ $( 1, 6,11, 2, 7,12, 3, 8,13, 4, 9,14, 5,10)$
14A-5 $14$ $1$ $14$ $13$ $( 1,10, 5,14, 9, 4,13, 8, 3,12, 7, 2,11, 6)$

Malle's constant $a(G)$:     $1/7$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 7A1 7A-1 7A2 7A-2 7A3 7A-3 14A1 14A-1 14A3 14A-3 14A5 14A-5
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 7A2 7A-2 7A-3 7A3 7A-1 7A1 7A1 7A-1 7A3 7A-3 7A-2 7A2
7 P 1A 2A 7A3 7A-3 7A-1 7A1 7A2 7A-2 14A3 14A-3 14A-5 14A5 14A1 14A-1
Type
14.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
14.2.1b R 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
14.2.1c1 C 1 1 ζ7−3 ζ7−1 ζ7 ζ7−2 ζ73 ζ72 ζ7−1 ζ72 ζ73 ζ7−2 ζ7 ζ7−3
14.2.1c2 C 1 1 ζ73 ζ7 ζ7−1 ζ72 ζ7−3 ζ7−2 ζ7 ζ7−2 ζ7−3 ζ72 ζ7−1 ζ73
14.2.1c3 C 1 1 ζ7−2 ζ7−3 ζ73 ζ7 ζ72 ζ7−1 ζ7−3 ζ7−1 ζ72 ζ7 ζ73 ζ7−2
14.2.1c4 C 1 1 ζ72 ζ73 ζ7−3 ζ7−1 ζ7−2 ζ7 ζ73 ζ7 ζ7−2 ζ7−1 ζ7−3 ζ72
14.2.1c5 C 1 1 ζ7−1 ζ72 ζ7−2 ζ7−3 ζ7 ζ73 ζ72 ζ73 ζ7 ζ7−3 ζ7−2 ζ7−1
14.2.1c6 C 1 1 ζ7 ζ7−2 ζ72 ζ73 ζ7−1 ζ7−3 ζ7−2 ζ7−3 ζ7−1 ζ73 ζ72 ζ7
14.2.1d1 C 1 −1 ζ7−3 ζ7−1 ζ7 ζ7−2 ζ73 ζ72 −ζ7−1 −ζ72 −ζ73 −ζ7−2 −ζ7 −ζ7−3
14.2.1d2 C 1 −1 ζ73 ζ7 ζ7−1 ζ72 ζ7−3 ζ7−2 −ζ7 −ζ7−2 −ζ7−3 −ζ72 −ζ7−1 −ζ73
14.2.1d3 C 1 −1 ζ7−2 ζ7−3 ζ73 ζ7 ζ72 ζ7−1 −ζ7−3 −ζ7−1 −ζ72 −ζ7 −ζ73 −ζ7−2
14.2.1d4 C 1 −1 ζ72 ζ73 ζ7−3 ζ7−1 ζ7−2 ζ7 −ζ73 −ζ7 −ζ7−2 −ζ7−1 −ζ7−3 −ζ72
14.2.1d5 C 1 −1 ζ7−1 ζ72 ζ7−2 ζ7−3 ζ7 ζ73 −ζ72 −ζ73 −ζ7 −ζ7−3 −ζ7−2 −ζ7−1
14.2.1d6 C 1 −1 ζ7 ζ7−2 ζ72 ζ73 ζ7−1 ζ7−3 −ζ7−2 −ζ7−3 −ζ7−1 −ζ73 −ζ72 −ζ7

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{14} + \left(-42 t^{6} - 42 t^{5} - 42 t^{4} - 42 t^{3} - 42 t^{2} - 42 t - 42\right) x^{12} + \left(231 t^{12} + 462 t^{11} + 1477 t^{10} + 2100 t^{9} + 2723 t^{8} + 3738 t^{7} + 4361 t^{6} + 4130 t^{5} + 3899 t^{4} + 2884 t^{3} + 2261 t^{2} + 1638 t + 623\right) x^{10} + \left(-560 t^{18} - 1680 t^{17} - 5320 t^{16} - 10500 t^{15} - 28196 t^{14} - 48412 t^{13} - 73059 t^{12} - 109277 t^{11} - 142660 t^{10} - 165858 t^{9} - 185976 t^{8} - 184492 t^{7} - 168707 t^{6} - 144060 t^{5} - 107282 t^{4} - 72779 t^{3} - 45941 t^{2} - 20643 t - 4431\right) x^{8} + \left(735 t^{24} + 2940 t^{23} + 7350 t^{22} + 14700 t^{21} + 44933 t^{20} + 98784 t^{19} + 250390 t^{18} + 478338 t^{17} + 804433 t^{16} + 1332114 t^{15} + 1991948 t^{14} + 2690492 t^{13} + 3425541 t^{12} + 3998694 t^{11} + 4349730 t^{10} + 4466448 t^{9} + 4206993 t^{8} + 3640798 t^{7} + 2928534 t^{6} + 2139634 t^{5} + 1414875 t^{4} + 835156 t^{3} + 390138 t^{2} + 117502 t + 16513\right) x^{6} + \left(-546 t^{30} - 2730 t^{29} - 4270 t^{28} - 1470 t^{27} - 8428 t^{26} - 35868 t^{25} - 125048 t^{24} - 297892 t^{23} - 750785 t^{22} - 1464827 t^{21} - 2796528 t^{20} - 5606090 t^{19} - 9994089 t^{18} - 15860761 t^{17} - 23528421 t^{16} - 32329332 t^{15} - 41411902 t^{14} - 49685559 t^{13} - 55258231 t^{12} - 57427706 t^{11} - 56236124 t^{10} - 51256618 t^{9} - 43197525 t^{8} - 33693401 t^{7} - 24122749 t^{6} - 15733557 t^{5} - 9148643 t^{4} - 4384373 t^{3} - 1522829 t^{2} - 325934 t - 31906\right) x^{4} + \left(217 t^{36} + 1302 t^{35} + 637 t^{34} - 9408 t^{33} - 18522 t^{32} - 2156 t^{31} + 52724 t^{30} + 297472 t^{29} + 916405 t^{28} + 1232252 t^{27} + 382347 t^{26} - 1519490 t^{25} - 3259137 t^{24} - 2846508 t^{23} + 1371524 t^{22} + 11142390 t^{21} + 30388820 t^{20} + 62408654 t^{19} + 107381981 t^{18} + 163149420 t^{17} + 223881000 t^{16} + 281421294 t^{15} + 328575247 t^{14} + 358521632 t^{13} + 366484769 t^{12} + 351222396 t^{11} + 313783309 t^{10} + 260040354 t^{9} + 200267907 t^{8} + 143108560 t^{7} + 93629347 t^{6} + 54506326 t^{5} + 26706323 t^{4} + 10101252 t^{3} + 2645118 t^{2} + 414274 t + 28784\right) x^{2} + \left(-36 t^{42} - 252 t^{41} + 168 t^{40} + 4620 t^{39} + 7532 t^{38} - 16044 t^{37} - 54201 t^{36} - 43821 t^{35} + 4053 t^{34} - 37443 t^{33} - 329742 t^{32} - 512176 t^{31} + 372554 t^{30} + 2425535 t^{29} + 4374502 t^{28} + 5507922 t^{27} + 7679343 t^{26} + 13661928 t^{25} + 20026160 t^{24} + 14854336 t^{23} - 12506095 t^{22} - 63221390 t^{21} - 134266594 t^{20} - 229560359 t^{19} - 356950734 t^{18} - 507887702 t^{17} - 651927591 t^{16} - 759720724 t^{15} - 822126747 t^{14} - 842090809 t^{13} - 815973914 t^{12} - 735878941 t^{11} - 610620430 t^{10} - 468684125 t^{9} - 336922663 t^{8} - 225932540 t^{7} - 136113761 t^{6} - 69067236 t^{5} - 27466572 t^{4} - 7972825 t^{3} - 1558571 t^{2} - 181293 t - 9409\right)$ Copy content Toggle raw display