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Properties

Label 30T29
Degree $30$
Order $120$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2\times A_5$

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Show commands: Magma

magma: G := TransitiveGroup(30, 29);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $29$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times A_5$
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:  (1,16,11,2,15,12)(3,25,30,4,26,29)(5,13,21,6,14,22)(7,27,9,8,28,10)(17,24,19,18,23,20), (1,19,28,18,5,2,20,27,17,6)(3,13,12,30,24,4,14,11,29,23)(7,22,15,10,25,8,21,16,9,26)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$60$:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 5: $A_5$

Degree 6: None

Degree 10: None

Degree 15: $A_5$

Low degree siblings

10T11, 12T75, 12T76, 20T31, 20T36, 24T203, 30T30, 40T61

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  120.35
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 5A1 5A2 6A 10A1 10A3
Size 1 1 15 15 20 12 12 20 12 12
2 P 1A 1A 1A 1A 3A 5A2 5A1 3A 5A1 5A2
3 P 1A 2A 2B 2C 1A 5A2 5A1 2A 10A3 10A1
5 P 1A 2A 2B 2C 3A 1A 1A 6A 2A 2A
Type
120.35.1a R 1 1 1 1 1 1 1 1 1 1
120.35.1b R 1 −1 −1 1 1 1 1 −1 −1 −1
120.35.3a1 R 3 3 −1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5
120.35.3a2 R 3 3 −1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52
120.35.3b1 R 3 −3 1 −1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5
120.35.3b2 R 3 −3 1 −1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52
120.35.4a R 4 4 0 0 1 −1 −1 1 −1 −1
120.35.4b R 4 −4 0 0 1 −1 −1 −1 1 1
120.35.5a R 5 5 1 1 −1 0 0 −1 0 0
120.35.5b R 5 −5 −1 1 −1 0 0 1 0 0

magma: CharacterTable(G);