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

$\card{(G/N)}$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

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{30}$ $1$ $1$ $0$ $()$
2A $2^{15}$ $3$ $2$ $15$ $( 1, 2)( 3,23)( 4,24)( 5,16)( 6,15)( 7, 8)( 9,30)(10,29)(11,21)(12,22)(13,14)(17,28)(18,27)(19,20)(25,26)$
2B $2^{12},1^{6}$ $5$ $2$ $12$ $( 1,20)( 2,19)( 3,28)( 4,27)( 7,13)( 8,14)( 9,22)(10,21)(11,29)(12,30)(17,23)(18,24)$
2C $2^{15}$ $15$ $2$ $15$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,29)(22,30)(23,28)(24,27)(25,26)$
3A $3^{10}$ $2$ $3$ $20$ $( 1,11,22)( 2,12,21)( 3,13,24)( 4,14,23)( 5,15,26)( 6,16,25)( 7,18,28)( 8,17,27)( 9,20,29)(10,19,30)$
4A1 $4^{6},2^{3}$ $5$ $4$ $21$ $( 1, 8,20,14)( 2, 7,19,13)( 3,21,28,10)( 4,22,27, 9)( 5, 6)(11,17,29,23)(12,18,30,24)(15,16)(25,26)$
4A-1 $4^{6},2^{3}$ $5$ $4$ $21$ $( 1,14,20, 8)( 2,13,19, 7)( 3,10,28,21)( 4, 9,27,22)( 5, 6)(11,23,29,17)(12,24,30,18)(15,16)(25,26)$
4B1 $4^{6},2^{2},1^{2}$ $15$ $4$ $20$ $( 1, 7,20,13)( 2, 8,19,14)( 3,11,28,29)( 4,12,27,30)( 5,15)( 6,16)( 9,24,22,18)(10,23,21,17)$
4B-1 $4^{6},2^{2},1^{2}$ $15$ $4$ $20$ $( 1,13,20, 7)( 2,14,19, 8)( 3,29,28,11)( 4,30,27,12)( 5,15)( 6,16)( 9,18,22,24)(10,17,21,23)$
5A $5^{6}$ $4$ $5$ $24$ $( 1,20, 7,26,13)( 2,19, 8,25,14)( 3,22, 9,28,15)( 4,21,10,27,16)( 5,24,11,29,18)( 6,23,12,30,17)$
6A $6^{4},3^{2}$ $10$ $6$ $24$ $( 1,24,22,13,11, 3)( 2,23,21,14,12, 4)( 5, 9,26,29,15,20)( 6,10,25,30,16,19)( 7,18,28)( 8,17,27)$
10A $10^{3}$ $12$ $10$ $27$ $( 1,19, 7,25,13, 2,20, 8,26,14)( 3,12, 9,17,15,23,22,30,28, 6)( 4,11,10,18,16,24,21,29,27, 5)$
12A1 $12^{2},6$ $10$ $12$ $27$ $( 1,12,22, 2,11,21)( 3, 8,29,16,13,17, 9,25,24,27,20, 6)( 4, 7,30,15,14,18,10,26,23,28,19, 5)$
12A-1 $12^{2},6$ $10$ $12$ $27$ $( 1,30,15, 8,11,10,26,17,22,19, 5,27)( 2,29,16, 7,12, 9,25,18,21,20, 6,28)( 3,14,24, 4,13,23)$
15A $15^{2}$ $8$ $15$ $28$ $( 1,18, 3,20, 5,22, 7,24, 9,26,11,28,13,29,15)( 2,17, 4,19, 6,21, 8,23,10,25,12,27,14,30,16)$

Malle's constant $a(G)$:     $1/12$

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:

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 5A 6A 10A 12A1 12A-1 15A
Size 1 3 5 15 2 5 5 15 15 4 10 12 10 10 8
2 P 1A 1A 1A 1A 3A 2B 2B 2B 2B 5A 3A 5A 6A 6A 15A
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 5A 2B 10A 4A1 4A-1 5A
5 P 1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 1A 6A 2A 12A1 12A-1 3A
Type
120.36.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
120.36.1b R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
120.36.1c R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
120.36.1d R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
120.36.1e1 C 1 1 1 1 1 i i i i 1 1 1 i i 1
120.36.1e2 C 1 1 1 1 1 i i i i 1 1 1 i i 1
120.36.1f1 C 1 1 1 1 1 i i i i 1 1 1 i i 1
120.36.1f2 C 1 1 1 1 1 i i i i 1 1 1 i i 1
120.36.2a R 2 0 2 0 1 2 2 0 0 2 1 0 1 1 1
120.36.2b R 2 0 2 0 1 2 2 0 0 2 1 0 1 1 1
120.36.2c1 C 2 0 2 0 1 2i 2i 0 0 2 1 0 i i 1
120.36.2c2 C 2 0 2 0 1 2i 2i 0 0 2 1 0 i i 1
120.36.4a R 4 4 0 0 4 0 0 0 0 1 0 1 0 0 1
120.36.4b R 4 4 0 0 4 0 0 0 0 1 0 1 0 0 1
120.36.8a R 8 0 0 0 4 0 0 0 0 2 0 0 0 0 1

magma: CharacterTable(G);