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Properties

Label 30T38
Degree $30$
Order $150$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^2:S_3$

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Show commands: Magma

magma: G := TransitiveGroup(30, 38);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $38$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^2:S_3$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,12,22)(2,11,21)(3,13,24)(4,14,23)(5,15,26)(6,16,25)(7,18,27)(8,17,28)(9,20,30)(10,19,29), (1,27,8,3,14,10,20,15,25,21)(2,28,7,4,13,9,19,16,26,22)(5,30,24,17,11,6,29,23,18,12)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: None

Degree 6: $S_3$

Degree 10: None

Degree 15: $(C_5^2 : C_3):C_2$

Low degree siblings

15T13, 15T14, 25T16, 30T37

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrder IndexRepresentative
1A $1^{30}$ $1$ $1$ $0$ $()$
2A $2^{15}$ $15$ $2$ $15$ $( 1,21)( 2,22)( 3,14)( 4,13)( 5, 6)( 7,28)( 8,27)( 9,19)(10,20)(11,12)(15,25)(16,26)(17,18)(23,24)(29,30)$
3A $3^{10}$ $50$ $3$ $20$ $( 1,12,22)( 2,11,21)( 3,13,24)( 4,14,23)( 5,15,26)( 6,16,25)( 7,18,27)( 8,17,28)( 9,20,30)(10,19,29)$
5A1 $5^{6}$ $3$ $5$ $24$ $( 1, 8,14,20,25)( 2, 7,13,19,26)( 3,15,27,10,21)( 4,16,28, 9,22)( 5,18,29,11,24)( 6,17,30,12,23)$
5A-1 $5^{6}$ $3$ $5$ $24$ $( 1,14,25, 8,20)( 2,13,26, 7,19)( 3,27,21,15,10)( 4,28,22,16, 9)( 5,29,24,18,11)( 6,30,23,17,12)$
5A2 $5^{6}$ $3$ $5$ $24$ $( 1,25,20,14, 8)( 2,26,19,13, 7)( 3,21,10,27,15)( 4,22, 9,28,16)( 5,24,11,29,18)( 6,23,12,30,17)$
5A-2 $5^{6}$ $3$ $5$ $24$ $( 1,20, 8,25,14)( 2,19, 7,26,13)( 3,10,15,21,27)( 4, 9,16,22,28)( 5,11,18,24,29)( 6,12,17,23,30)$
5B1 $5^{4},1^{10}$ $6$ $5$ $16$ $( 1,20, 8,25,14)( 2,19, 7,26,13)( 3,15,27,10,21)( 4,16,28, 9,22)$
5B2 $5^{4},1^{10}$ $6$ $5$ $16$ $( 3,10,15,21,27)( 4, 9,16,22,28)( 5,29,24,18,11)( 6,30,23,17,12)$
10A1 $10^{3}$ $15$ $10$ $27$ $( 1,21, 8,27,14, 3,20,10,25,15)( 2,22, 7,28,13, 4,19, 9,26,16)( 5,30,24,17,11, 6,29,23,18,12)$
10A-1 $10^{3}$ $15$ $10$ $27$ $( 1,21,25,15,20,10,14, 3, 8,27)( 2,22,26,16,19, 9,13, 4, 7,28)( 5,12,18,23,29, 6,11,17,24,30)$
10A3 $10^{3}$ $15$ $10$ $27$ $( 1,21,20,10, 8,27,25,15,14, 3)( 2,22,19, 9, 7,28,26,16,13, 4)( 5,17,29,12,24, 6,18,30,11,23)$
10A-3 $10^{3}$ $15$ $10$ $27$ $( 1,21,14, 3,25,15, 8,27,20,10)( 2,22,13, 4,26,16, 7,28,19, 9)( 5,23,11,30,18, 6,24,12,29,17)$

Malle's constant $a(G)$:     $1/15$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $150=2 \cdot 3 \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  150.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 5A1 5A-1 5A2 5A-2 5B1 5B2 10A1 10A-1 10A3 10A-3
Size 1 15 50 3 3 3 3 6 6 15 15 15 15
2 P 1A 1A 3A 5A-1 5A-2 5A1 5A2 5B2 5B1 5A1 5A-1 5A-2 5A2
3 P 1A 2A 1A 5A1 5A2 5A-1 5A-2 5B2 5B1 10A3 10A-3 10A-1 10A1
5 P 1A 2A 3A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A
Type
150.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
150.5.1b R 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
150.5.2a R 2 0 −1 2 2 2 2 2 2 0 0 0 0
150.5.3a1 C 3 1 0 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ52 ζ5−2 ζ5 ζ5−1
150.5.3a2 C 3 1 0 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2 ζ52 ζ5−1 ζ5
150.5.3a3 C 3 1 0 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5 ζ5−1 ζ5−2 ζ52
150.5.3a4 C 3 1 0 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1 ζ5 ζ52 ζ5−2
150.5.3b1 C 3 −1 0 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1
150.5.3b2 C 3 −1 0 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5
150.5.3b3 C 3 −1 0 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52
150.5.3b4 C 3 −1 0 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2
150.5.6a1 R 6 0 0 −2ζ5−1−2ζ5 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 −2ζ5−2−2ζ52 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 0 0 0 0
150.5.6a2 R 6 0 0 −2ζ5−2−2ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 −2ζ5−1−2ζ5 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 0 0 0 0

magma: CharacterTable(G);