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Properties

Label 30T40
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, 40);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
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:  (1,19,7,25,14)(2,20,8,26,13)(5,17,29,11,23)(6,18,30,12,24), (1,16,5,2,15,6)(3,24,19,4,23,20)(7,22,11,8,21,12)(9,30,25,10,29,26)(13,27,18,14,28,17)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$75$:  $C_5^2 : C_3$

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: $C_5^2 : C_3$

Low degree siblings

30T40

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

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:  150.7
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 5A1 5A-1 5A2 5A-2 5B1 5B-1 5B2 5B-2 6A1 6A-1 10A1 10A-1 10A3 10A-3 10B1 10B-1 10B3 10B-3
Size 1 1 25 25 3 3 3 3 3 3 3 3 25 25 3 3 3 3 3 3 3 3
2 P 1A 1A 3A-1 3A1 5B1 5A1 5A2 5B-1 5A-2 5B2 5B-2 5A-1 3A1 3A-1 5A1 5B-2 5A2 5B2 5A-1 5B-1 5B1 5A-2
3 P 1A 2A 1A 1A 5B-1 5A-1 5A-2 5B1 5A2 5B-2 5B2 5A1 2A 2A 10A3 10B-1 10A1 10B1 10A-3 10B-3 10B3 10A-1
5 P 1A 2A 3A-1 3A1 1A 1A 1A 1A 1A 1A 1A 1A 6A-1 6A1 2A 2A 2A 2A 2A 2A 2A 2A
Type
150.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
150.7.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
150.7.1c1 C 1 1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ3−1 1 1 1 1 1 1 1 1
150.7.1c2 C 1 1 ζ3 ζ3−1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 1 1 1 1 1 1 1 1
150.7.1d1 C 1 −1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 −ζ3 −ζ3−1 −1 −1 −1 −1 −1 −1 −1 −1
150.7.1d2 C 1 −1 ζ3 ζ3−1 1 1 1 1 1 1 1 1 −ζ3−1 −ζ3 −1 −1 −1 −1 −1 −1 −1 −1
150.7.3a1 C 3 3 0 0 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 0 0 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
150.7.3a2 C 3 3 0 0 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 0 0 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
150.7.3a3 C 3 3 0 0 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 0 0 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
150.7.3a4 C 3 3 0 0 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 0 0 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
150.7.3b1 C 3 3 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52
150.7.3b2 C 3 3 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52
150.7.3b3 C 3 3 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52
150.7.3b4 C 3 3 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5
150.7.3c1 C 3 −3 0 0 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 0 0 2ζ5−2+2+2ζ5+ζ52 −ζ5−2−2ζ5 −ζ5−2ζ52 −ζ5−2+1+ζ5+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
150.7.3c2 C 3 −3 0 0 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 0 0 −ζ5−2−2ζ5 2ζ5−2+2+2ζ5+ζ52 −ζ5−2+1+ζ5+ζ52 −ζ5−2ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
150.7.3c3 C 3 −3 0 0 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 0 0 −ζ5−2ζ52 −ζ5−2+1+ζ5+ζ52 −ζ5−2−2ζ5 2ζ5−2+2+2ζ5+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
150.7.3c4 C 3 −3 0 0 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 0 0 −ζ5−2+1+ζ5+ζ52 −ζ5−2ζ52 2ζ5−2+2+2ζ5+ζ52 −ζ5−2−2ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
150.7.3d1 C 3 −3 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 0 0 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 2ζ5−2+2+2ζ5+ζ52 −ζ5−2−2ζ5 −ζ5−2ζ52 −ζ5−2+1+ζ5+ζ52
150.7.3d2 C 3 −3 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 0 0 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 −ζ5−2−2ζ5 2ζ5−2+2+2ζ5+ζ52 −ζ5−2+1+ζ5+ζ52 −ζ5−2ζ52
150.7.3d3 C 3 −3 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5−2+2ζ5 ζ5−2−1−ζ5−ζ52 ζ5+2ζ52 0 0 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 −ζ5−2ζ52 −ζ5−2+1+ζ5+ζ52 −ζ5−2−2ζ5 2ζ5−2+2+2ζ5+ζ52
150.7.3d4 C 3 −3 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 ζ5−2+2ζ5 −2ζ5−2−2−2ζ5−ζ52 ζ5+2ζ52 ζ5−2−1−ζ5−ζ52 0 0 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 −ζ5−2+1+ζ5+ζ52 −ζ5−2ζ52 2ζ5−2+2+2ζ5+ζ52 −ζ5−2−2ζ5

magma: CharacterTable(G);