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Properties

Label 30T30
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, 30);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $30$
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);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,21,26,20,3,2,22,25,19,4)(5,27,29,7,11,6,28,30,8,12)(9,17,16,13,24,10,18,15,14,23), (1,25,5,2,26,6)(3,18,8,4,17,7)(9,19,30,10,20,29)(11,27,13,12,28,14)(15,24,22,16,23,21)
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: $C_2$

Degree 3: None

Degree 5: $A_5$

Degree 6: None

Degree 10: $A_5\times C_2$

Degree 15: $A_5$

Low degree siblings

10T11, 12T75, 12T76, 20T31, 20T36, 24T203, 30T29, 40T61

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:    
      2  3  3  3  3  1  1  1   1   1  1
      3  1  .  1  .  .  .  1   .   .  1
      5  1  .  1  .  1  1  .   1   1  .

        1a 2a 2b 2c 5a 5b 3a 10a 10b 6a
     2P 1a 1a 1a 1a 5b 5a 3a  5b  5a 3a
     3P 1a 2a 2b 2c 5b 5a 1a 10b 10a 2b
     5P 1a 2a 2b 2c 1a 1a 3a  2b  2b 6a
     7P 1a 2a 2b 2c 5b 5a 3a 10b 10a 6a

X.1      1  1  1  1  1  1  1   1   1  1
X.2      1  1 -1 -1  1  1  1  -1  -1 -1
X.3      3 -1 -3  1  A *A  .  -A -*A  .
X.4      3 -1 -3  1 *A  A  . -*A  -A  .
X.5      3 -1  3 -1  A *A  .   A  *A  .
X.6      3 -1  3 -1 *A  A  .  *A   A  .
X.7      4  .  4  . -1 -1  1  -1  -1  1
X.8      4  . -4  . -1 -1  1   1   1 -1
X.9      5  1  5  1  .  . -1   .   . -1
X.10     5  1 -5 -1  .  . -1   .   .  1

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5

magma: CharacterTable(G);