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Properties

Label 30T50
30T50 1 18 1->18 2 17 2->17 3 12 3->12 4 11 4->11 5 29 5->29 6 30 6->30 7 7->6 8 7->8 8->5 9 10 9->10 20 9->20 19 10->19 11->9 11->12 12->10 13 14 13->14 23 13->23 24 14->24 15 15->3 16 15->16 16->4 17->18 22 17->22 21 18->21 19->7 19->20 20->8 21->22 25 21->25 26 22->26 23->1 24->2 25->15 26->16 27 27->13 28 28->14 29->28 30->27
Degree $30$
Order $240$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_{16}$

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Copy content magma:G := TransitiveGroup(30, 50);
 

Group invariants

Abstract group:  $F_{16}$
Copy content magma:IdentifyGroup(G);
 
Order:  $240=2^{4} \cdot 3 \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $30$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $50$
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:  $(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$, $(1,18,21,25,15,3,12,10,19,7,6,30,27,13,23)(2,17,22,26,16,4,11,9,20,8,5,29,28,14,24)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$
$5$:  $C_5$
$15$:  $C_{15}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 5: $C_5$

Degree 6: None

Degree 10: None

Degree 15: $C_{15}$

Low degree siblings

16T447, 20T67

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{30}$ $1$ $1$ $0$ $()$
2A $2^{8},1^{14}$ $15$ $2$ $8$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(15,16)(17,18)(27,28)(29,30)$
3A1 $3^{10}$ $16$ $3$ $20$ $( 1, 3, 6)( 2, 4, 5)( 7,23,16)( 8,24,15)( 9,28,21)(10,27,22)(11,29,18)(12,30,17)(13,25,20)(14,26,19)$
3A-1 $3^{10}$ $16$ $3$ $20$ $( 1, 6, 3)( 2, 5, 4)( 7,16,23)( 8,15,24)( 9,21,28)(10,22,27)(11,18,29)(12,17,30)(13,20,25)(14,19,26)$
5A1 $5^{6}$ $16$ $5$ $24$ $( 1, 8,26,28,11)( 2, 7,25,27,12)( 3,24,19,21,29)( 4,23,20,22,30)( 5,16,13,10,17)( 6,15,14, 9,18)$
5A-1 $5^{6}$ $16$ $5$ $24$ $( 1,11,28,26, 8)( 2,12,27,25, 7)( 3,29,21,19,24)( 4,30,22,20,23)( 5,17,10,13,16)( 6,18, 9,14,15)$
5A2 $5^{6}$ $16$ $5$ $24$ $( 1,26,11, 8,28)( 2,25,12, 7,27)( 3,19,29,24,21)( 4,20,30,23,22)( 5,13,17,16,10)( 6,14,18,15, 9)$
5A-2 $5^{6}$ $16$ $5$ $24$ $( 1,28, 8,11,26)( 2,27, 7,12,25)( 3,21,24,29,19)( 4,22,23,30,20)( 5,10,16,17,13)( 6, 9,15,18,14)$
15A1 $15^{2}$ $16$ $15$ $28$ $( 1,14,29, 8, 9, 3,26,18,24,28, 6,19,11,15,21)( 2,13,30, 7,10, 4,25,17,23,27, 5,20,12,16,22)$
15A-1 $15^{2}$ $16$ $15$ $28$ $( 1,21,15,11,19, 6,28,24,18,26, 3, 9, 8,29,14)( 2,22,16,12,20, 5,27,23,17,25, 4,10, 7,30,13)$
15A2 $15^{2}$ $16$ $15$ $28$ $( 1,29, 9,26,24, 6,11,21,14, 8, 3,18,28,19,15)( 2,30,10,25,23, 5,12,22,13, 7, 4,17,27,20,16)$
15A-2 $15^{2}$ $16$ $15$ $28$ $( 1,15,19,28,18, 3, 8,14,21,11, 6,24,26, 9,29)( 2,16,20,27,17, 4, 7,13,22,12, 5,23,25,10,30)$
15A4 $15^{2}$ $16$ $15$ $28$ $( 1, 9,24,11,14, 3,28,15,29,26, 6,21, 8,18,19)( 2,10,23,12,13, 4,27,16,30,25, 5,22, 7,17,20)$
15A-4 $15^{2}$ $16$ $15$ $28$ $( 1,19,18, 8,21, 6,26,29,15,28, 3,14,11,24, 9)( 2,20,17, 7,22, 5,25,30,16,27, 4,13,12,23,10)$
15A7 $15^{2}$ $16$ $15$ $28$ $( 1,18,21,26,15, 3,11, 9,19, 8, 6,29,28,14,24)( 2,17,22,25,16, 4,12,10,20, 7, 5,30,27,13,23)$
15A-7 $15^{2}$ $16$ $15$ $28$ $( 1,24,14,28,29, 6, 8,19, 9,11, 3,15,26,21,18)( 2,23,13,27,30, 5, 7,20,10,12, 4,16,25,22,17)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 5A1 5A-1 5A2 5A-2 15A1 15A-1 15A2 15A-2 15A4 15A-4 15A7 15A-7
Size 1 15 16 16 16 16 16 16 16 16 16 16 16 16 16 16
2 P 1A 1A 3A-1 3A1 5A2 5A-2 5A-1 5A1 15A2 15A-2 15A4 15A-4 15A-7 15A7 15A-1 15A1
3 P 1A 2A 1A 1A 5A-2 5A2 5A1 5A-1 5A1 5A-1 5A2 5A-2 5A-1 5A1 5A2 5A-2
5 P 1A 2A 3A-1 3A1 1A 1A 1A 1A 3A1 3A-1 3A-1 3A1 3A1 3A-1 3A1 3A-1
Type
240.191.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
240.191.1b1 C 1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3
240.191.1b2 C 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1
240.191.1c1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−2
240.191.1c2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ52
240.191.1c3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ5−1
240.191.1c4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5
240.191.1d1 C 1 1 ζ15−5 ζ155 ζ15−6 ζ156 ζ153 ζ15−3 ζ152 ζ15 ζ154 ζ15−4 ζ157 ζ15−2 ζ15−7 ζ15−1
240.191.1d2 C 1 1 ζ155 ζ15−5 ζ156 ζ15−6 ζ15−3 ζ153 ζ15−2 ζ15−1 ζ15−4 ζ154 ζ15−7 ζ152 ζ157 ζ15
240.191.1d3 C 1 1 ζ15−5 ζ155 ζ156 ζ15−6 ζ15−3 ζ153 ζ15−7 ζ154 ζ15 ζ15−1 ζ15−2 ζ157 ζ152 ζ15−4
240.191.1d4 C 1 1 ζ155 ζ15−5 ζ15−6 ζ156 ζ153 ζ15−3 ζ157 ζ15−4 ζ15−1 ζ15 ζ152 ζ15−7 ζ15−2 ζ154
240.191.1d5 C 1 1 ζ15−5 ζ155 ζ15−3 ζ153 ζ15−6 ζ156 ζ15−4 ζ15−2 ζ157 ζ15−7 ζ15 ζ154 ζ15−1 ζ152
240.191.1d6 C 1 1 ζ155 ζ15−5 ζ153 ζ15−3 ζ156 ζ15−6 ζ154 ζ152 ζ15−7 ζ157 ζ15−1 ζ15−4 ζ15 ζ15−2
240.191.1d7 C 1 1 ζ15−5 ζ155 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−1 ζ157 ζ15−2 ζ152 ζ154 ζ15 ζ15−4 ζ15−7
240.191.1d8 C 1 1 ζ155 ζ15−5 ζ15−3 ζ153 ζ15−6 ζ156 ζ15 ζ15−7 ζ152 ζ15−2 ζ15−4 ζ15−1 ζ154 ζ157
240.191.15a R 15 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed