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Properties

Label 30T35
Degree $30$
Order $150$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^2:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(30, 35);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $35$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^2:C_6$
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:  (3,10,15,21,27)(4,9,16,22,28)(5,29,23,17,11)(6,30,24,18,12), (1,18,22,8,11,27)(2,17,21,7,12,28)(3,25,24,16,14,5)(4,26,23,15,13,6)(9,20,29,10,19,30)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 5: None

Degree 6: $C_6$

Degree 10: None

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

Low degree siblings

15T12 x 2, 25T15, 30T35

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  150.6
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 5A1 5A2 5B1 5B2 6A1 6A-1
Size 1 25 25 25 6 6 6 6 25 25
2 P 1A 1A 3A-1 3A1 5A2 5A1 5B2 5B1 3A1 3A-1
3 P 1A 2A 1A 1A 5A2 5A1 5B2 5B1 2A 2A
5 P 1A 2A 3A-1 3A1 1A 1A 1A 1A 6A-1 6A1
Type
150.6.1a R 1 1 1 1 1 1 1 1 1 1
150.6.1b R 1 −1 1 1 1 1 1 1 −1 −1
150.6.1c1 C 1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1
150.6.1c2 C 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3
150.6.1d1 C 1 −1 ζ3−1 ζ3 1 1 1 1 −ζ3 −ζ3−1
150.6.1d2 C 1 −1 ζ3 ζ3−1 1 1 1 1 −ζ3−1 −ζ3
150.6.6a1 R 6 0 0 0 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0
150.6.6a2 R 6 0 0 0 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0
150.6.6b1 R 6 0 0 0 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 0 0
150.6.6b2 R 6 0 0 0 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 0 0

magma: CharacterTable(G);