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Properties

Label 30T11
Degree $30$
Order $60$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(30, 11);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $11$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5\times A_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20), (1,20,29,10,17,28,7,15,25,5,14,23,4,11,21)(2,19,30,9,18,27,8,16,26,6,13,24,3,12,22)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$5$:  $C_5$
$12$:  $A_4$
$15$:  $C_{15}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 5: $C_5$

Degree 6: $A_4$

Degree 10: None

Degree 15: $C_{15}$

Low degree siblings

20T14

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  60.9
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 15A1 15A-1 15A2 15A-2 15A4 15A-4 15A7 15A-7
Size 1 3 4 4 1 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4
2 P 1A 1A 3A-1 3A1 5A2 5A-2 5A-1 5A1 5A1 5A2 5A-1 5A-2 15A2 15A7 15A-4 15A4 15A-7 15A-2 15A-1 15A1
3 P 1A 2A 1A 1A 5A-2 5A2 5A1 5A-1 10A3 10A1 10A-3 10A-1 5A2 5A2 5A1 5A-1 5A-2 5A-2 5A-1 5A1
5 P 1A 2A 3A-1 3A1 1A 1A 1A 1A 2A 2A 2A 2A 3A1 3A-1 3A1 3A-1 3A1 3A-1 3A1 3A-1
Type
60.9.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
60.9.1b1 C 1 1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1
60.9.1b2 C 1 1 ζ3 ζ3−1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3
60.9.1c1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5
60.9.1c2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1
60.9.1c3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2
60.9.1c4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52
60.9.1d1 C 1 1 ζ15−5 ζ155 ζ15−6 ζ156 ζ153 ζ15−3 ζ15−3 ζ153 ζ156 ζ15−6 ζ15−4 ζ154 ζ157 ζ15−7 ζ15−1 ζ15 ζ152 ζ15−2
60.9.1d2 C 1 1 ζ155 ζ15−5 ζ156 ζ15−6 ζ15−3 ζ153 ζ153 ζ15−3 ζ15−6 ζ156 ζ154 ζ15−4 ζ15−7 ζ157 ζ15 ζ15−1 ζ15−2 ζ152
60.9.1d3 C 1 1 ζ15−5 ζ155 ζ156 ζ15−6 ζ15−3 ζ153 ζ153 ζ15−3 ζ15−6 ζ156 ζ15−1 ζ15 ζ15−2 ζ152 ζ15−4 ζ154 ζ15−7 ζ157
60.9.1d4 C 1 1 ζ155 ζ15−5 ζ15−6 ζ156 ζ153 ζ15−3 ζ15−3 ζ153 ζ156 ζ15−6 ζ15 ζ15−1 ζ152 ζ15−2 ζ154 ζ15−4 ζ157 ζ15−7
60.9.1d5 C 1 1 ζ15−5 ζ155 ζ15−3 ζ153 ζ15−6 ζ156 ζ156 ζ15−6 ζ153 ζ15−3 ζ15−7 ζ157 ζ15 ζ15−1 ζ152 ζ15−2 ζ15−4 ζ154
60.9.1d6 C 1 1 ζ155 ζ15−5 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−6 ζ156 ζ15−3 ζ153 ζ157 ζ15−7 ζ15−1 ζ15 ζ15−2 ζ152 ζ154 ζ15−4
60.9.1d7 C 1 1 ζ15−5 ζ155 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−6 ζ156 ζ15−3 ζ153 ζ152 ζ15−2 ζ154 ζ15−4 ζ15−7 ζ157 ζ15−1 ζ15
60.9.1d8 C 1 1 ζ155 ζ15−5 ζ15−3 ζ153 ζ15−6 ζ156 ζ156 ζ15−6 ζ153 ζ15−3 ζ15−2 ζ152 ζ15−4 ζ154 ζ157 ζ15−7 ζ15 ζ15−1
60.9.3a R 3 −1 0 0 3 3 3 3 −1 −1 −1 −1 0 0 0 0 0 0 0 0
60.9.3b1 C 3 −1 0 0 3ζ5−2 3ζ52 3ζ5 3ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 0 0 0 0 0 0 0 0
60.9.3b2 C 3 −1 0 0 3ζ52 3ζ5−2 3ζ5−1 3ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 0 0 0 0 0 0 0 0
60.9.3b3 C 3 −1 0 0 3ζ5−1 3ζ5 3ζ5−2 3ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 0 0 0 0 0 0 0 0
60.9.3b4 C 3 −1 0 0 3ζ5 3ζ5−1 3ζ52 3ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 0 0 0 0 0 0 0 0

magma: CharacterTable(G);