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Properties

Label 30T23
Degree $30$
Order $120$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times F_5$

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Show commands: Magma

magma: G := TransitiveGroup(30, 23);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $23$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times F_5$
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:  (1,7,20,13)(2,8,19,14)(3,11,28,29)(4,12,27,30)(5,15)(6,16)(9,24,22,18)(10,23,21,17), (1,21,11,2,22,12)(3,6,20,27,24,25,9,17,13,16,29,8)(4,5,19,28,23,26,10,18,14,15,30,7)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$20$:  $F_5$
$24$:  $S_3 \times C_4$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $F_5$

Degree 6: $D_{6}$

Degree 10: $F_{5}\times C_2$

Degree 15: $F_5 \times S_3$

Low degree siblings

15T11, 30T24, 30T32

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

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:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  120.36
magma: IdentifyGroup(G);
 
Character table:    
      2  3  3  3  3  3  3  3  3   .  2   2   1   2  1  2
      3  1  .  1  .  1  .  1  .   1  1   1   .   1  1  1
      5  1  .  .  .  .  1  .  .   1  .   .   1   .  1  1

        1a 4a 2a 4b 4c 2b 4d 2c 15a 6a 12a 10a 12b 5a 3a
     2P 1a 2a 1a 2a 2a 1a 2a 1a 15a 3a  6a  5a  6a 5a 3a
     3P 1a 4b 2a 4a 4d 2b 4c 2c  5a 2a  4d 10a  4c 5a 1a
     5P 1a 4a 2a 4b 4c 2b 4d 2c  3a 6a 12a  2b 12b 1a 3a
     7P 1a 4b 2a 4a 4d 2b 4c 2c 15a 6a 12b 10a 12a 5a 3a
    11P 1a 4b 2a 4a 4d 2b 4c 2c 15a 6a 12b 10a 12a 5a 3a
    13P 1a 4a 2a 4b 4c 2b 4d 2c 15a 6a 12a 10a 12b 5a 3a

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

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i

magma: CharacterTable(G);