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Properties

Label 30T2
Degree $30$
Order $30$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times S_3$

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Show commands: Magma

magma: G := TransitiveGroup(30, 2);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5\times S_3$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $30$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,22,12)(2,21,11)(3,24,14)(4,23,13)(5,26,16)(6,25,15)(7,28,18)(8,27,17)(9,30,19)(10,29,20), (1,27,14,9,26,21,7,4,20,15)(2,28,13,10,25,22,8,3,19,16)(5,11,18,23,29,6,12,17,24,30)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $C_5$

Degree 6: $S_3$

Degree 10: $C_{10}$

Degree 15: $S_3 \times C_5$

Low degree siblings

15T4

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $30=2 \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  30.1
magma: IdentifyGroup(G);
 
Character table:    
      2  1  1   .   1   .   1  1   1   1   .  .  1   .  1  1
      3  1  .   1   .   1   .  1   .   .   1  1  1   1  1  1
      5  1  1   1   1   1   1  1   1   1   1  1  1   1  1  1

        1a 2a 15a 10a 15b 10b 5a 10c 10d 15c 3a 5b 15d 5c 5d
     2P 1a 1a 15b  5d 15c  5c 5b  5b  5a 15d 3a 5d 15a 5a 5c
     3P 1a 2a  5a 10c  5b 10a 5c 10d 10b  5d 1a 5a  5c 5d 5b
     5P 1a 2a  3a  2a  3a  2a 1a  2a  2a  3a 3a 1a  3a 1a 1a
     7P 1a 2a 15b 10b 15c 10d 5b 10a 10c 15d 3a 5d 15a 5a 5c
    11P 1a 2a 15a 10a 15b 10b 5a 10c 10d 15c 3a 5b 15d 5c 5d
    13P 1a 2a 15d 10c 15a 10a 5c 10d 10b 15b 3a 5a 15c 5d 5b

X.1      1  1   1   1   1   1  1   1   1   1  1  1   1  1  1
X.2      1 -1   1  -1   1  -1  1  -1  -1   1  1  1   1  1  1
X.3      1 -1   A  -A   B  -B /B -/B -/A  /A  1  A  /B /A  B
X.4      1 -1   B  -B  /A -/A  A  -A -/B  /B  1  B   A /B /A
X.5      1 -1  /B -/B   A  -A /A -/A  -B   B  1 /B  /A  B  A
X.6      1 -1  /A -/A  /B -/B  B  -B  -A   A  1 /A   B  A /B
X.7      1  1   A   A   B   B /B  /B  /A  /A  1  A  /B /A  B
X.8      1  1   B   B  /A  /A  A   A  /B  /B  1  B   A /B /A
X.9      1  1  /B  /B   A   A /A  /A   B   B  1 /B  /A  B  A
X.10     1  1  /A  /A  /B  /B  B   B   A   A  1 /A   B  A /B
X.11     2  .  -1   .  -1   .  2   .   .  -1 -1  2  -1  2  2
X.12     2  . -/B   .  -A   .  C   .   .  -B -1 /D -/A  D /C
X.13     2  . -/A   . -/B   .  D   .   .  -A -1  C  -B /C /D
X.14     2  .  -A   .  -B   . /D   .   . -/A -1 /C -/B  C  D
X.15     2  .  -B   . -/A   . /C   .   . -/B -1  D  -A /D  C

A = E(5)^4
B = E(5)^3
C = 2*E(5)
D = 2*E(5)^3

magma: CharacterTable(G);