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Properties

Label 30T50
Degree $30$
Order $240$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_{16}$

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Show commands: Magma

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

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $50$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_{16}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
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)
magma: Generators(G);
 

Low degree resolvents

|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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7,23,15)( 8,24,16)( 9,28,21)(10,27,22)(11,30,17) (12,29,18)(13,26,20)(14,25,19)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7,15,23)( 8,16,24)( 9,21,28)(10,22,27)(11,17,30) (12,18,29)(13,20,26)(14,19,25)$
$ 5, 5, 5, 5, 5, 5 $ $16$ $5$ $( 1, 7,25,27,12)( 2, 8,26,28,11)( 3,23,19,21,30)( 4,24,20,22,29) ( 5,16,14, 9,17)( 6,15,13,10,18)$
$ 15, 15 $ $16$ $15$ $( 1, 9,24,11,13, 3,27,15,29,25, 5,21, 7,17,20)( 2,10,23,12,14, 4,28,16,30,26, 6,22, 8,18,19)$
$ 5, 5, 5, 5, 5, 5 $ $16$ $5$ $( 1,11,27,25, 7)( 2,12,28,26, 8)( 3,29,21,20,24)( 4,30,22,19,23) ( 5,17, 9,13,15)( 6,18,10,14,16)$
$ 15, 15 $ $16$ $15$ $( 1,13,29, 7, 9, 3,25,17,24,27, 5,20,11,15,21)( 2,14,30, 8,10, 4,26,18,23,28, 6,19,12,16,22)$
$ 15, 15 $ $16$ $15$ $( 1,15,19,27,18, 3, 7,13,21,12, 6,23,25,10,30)( 2,16,20,28,17, 4, 8,14,22,11, 5,24,26, 9,29)$
$ 15, 15 $ $16$ $15$ $( 1,17,21,25,15, 3,11, 9,20, 7, 5,29,27,13,24)( 2,18,22,26,16, 4,12,10,19, 8, 6,30,28,14,23)$
$ 15, 15 $ $16$ $15$ $( 1,19,18, 7,21, 6,25,30,15,27, 3,13,12,23,10)( 2,20,17, 8,22, 5,26,29,16,28, 4,14,11,24, 9)$
$ 15, 15 $ $16$ $15$ $( 1,21,15,11,20, 5,27,24,17,25, 3, 9, 7,29,13)( 2,22,16,12,19, 6,28,23,18,26, 4,10, 8,30,14)$
$ 15, 15 $ $16$ $15$ $( 1,23,13,27,30, 6, 7,19,10,12, 3,15,25,21,18)( 2,24,14,28,29, 5, 8,20, 9,11, 4,16,26,22,17)$
$ 5, 5, 5, 5, 5, 5 $ $16$ $5$ $( 1,25,11, 8,28)( 2,26,12, 7,27)( 3,19,29,23,22)( 4,20,30,24,21) ( 5,14,17,15, 9)( 6,13,18,16,10)$
$ 5, 5, 5, 5, 5, 5 $ $16$ $5$ $( 1,27, 7,11,25)( 2,28, 8,12,26)( 3,21,24,29,20)( 4,22,23,30,19) ( 5, 9,15,17,13)( 6,10,16,18,14)$
$ 15, 15 $ $16$ $15$ $( 1,29, 9,25,24, 5,11,21,13, 7, 3,17,27,20,15)( 2,30,10,26,23, 6,12,22,14, 8, 4,18,28,19,16)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $240=2^{4} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  240.191
magma: IdentifyGroup(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 15A-7 15A4 15A1 15A-1 15A-2 15A7 15A2 15A-4
3 P 1A 2A 1A 1A 5A-2 5A2 5A1 5A-1 5A-1 5A2 5A-2 5A2 5A-1 5A1 5A1 5A-2
5 P 1A 2A 3A-1 3A1 1A 1A 1A 1A 3A1 3A-1 3A-1 3A1 3A-1 3A-1 3A1 3A1
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 ζ3−1 ζ3 ζ3−1
240.191.1b2 C 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
240.191.1c1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2
240.191.1c2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52
240.191.1c3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1
240.191.1c4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5
240.191.1d1 C 1 1 ζ15−5 ζ155 ζ15−6 ζ156 ζ153 ζ15−3 ζ15−7 ζ157 ζ15 ζ15−1 ζ152 ζ15−2 ζ15−4 ζ154
240.191.1d2 C 1 1 ζ155 ζ15−5 ζ156 ζ15−6 ζ15−3 ζ153 ζ157 ζ15−7 ζ15−1 ζ15 ζ15−2 ζ152 ζ154 ζ15−4
240.191.1d3 C 1 1 ζ15−5 ζ155 ζ156 ζ15−6 ζ15−3 ζ153 ζ152 ζ15−2 ζ154 ζ15−4 ζ15−7 ζ157 ζ15−1 ζ15
240.191.1d4 C 1 1 ζ155 ζ15−5 ζ15−6 ζ156 ζ153 ζ15−3 ζ15−2 ζ152 ζ15−4 ζ154 ζ157 ζ15−7 ζ15 ζ15−1
240.191.1d5 C 1 1 ζ15−5 ζ155 ζ15−3 ζ153 ζ15−6 ζ156 ζ15−1 ζ15 ζ15−2 ζ152 ζ15−4 ζ154 ζ15−7 ζ157
240.191.1d6 C 1 1 ζ155 ζ15−5 ζ153 ζ15−3 ζ156 ζ15−6 ζ15 ζ15−1 ζ152 ζ15−2 ζ154 ζ15−4 ζ157 ζ15−7
240.191.1d7 C 1 1 ζ15−5 ζ155 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−4 ζ154 ζ157 ζ15−7 ζ15−1 ζ15 ζ152 ζ15−2
240.191.1d8 C 1 1 ζ155 ζ15−5 ζ15−3 ζ153 ζ15−6 ζ156 ζ154 ζ15−4 ζ15−7 ζ157 ζ15 ζ15−1 ζ15−2 ζ152
240.191.15a R 15 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);