↑

Properties

Label 30T19
Degree $30$
Order $120$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:S_4$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(30, 19);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:S_4$
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:  (1,29,17,8,25,14,3,22,20,10,27,16,5,23,12)(2,30,18,7,26,13,4,21,19,9,28,15,6,24,11), (1,2)(3,9)(4,10)(5,7)(6,8)(11,29,12,30)(13,27,14,28)(15,25,16,26)(17,24,18,23)(19,22,20,21)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$10$:  $D_{5}$
$24$:  $S_4$
$30$:  $D_{15}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 6: $S_4$

Degree 10: None

Degree 15: $D_{15}$

Low degree siblings

20T33, 30T31, 40T63

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

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:  120.38
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 4A 5A1 5A2 10A1 10A3 15A1 15A2 15A4 15A7
Size 1 3 30 8 30 2 2 6 6 8 8 8 8
2 P 1A 1A 1A 3A 2A 5A2 5A1 5A2 5A1 15A7 15A1 15A2 15A4
3 P 1A 2A 2B 1A 4A 5A2 5A1 10A3 10A1 5A1 5A2 5A1 5A2
5 P 1A 2A 2B 3A 4A 1A 1A 2A 2A 3A 3A 3A 3A
Type
120.38.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
120.38.1b R 1 1 −1 1 −1 1 1 1 1 1 1 1 1
120.38.2a R 2 2 0 −1 0 2 2 2 2 −1 −1 −1 −1
120.38.2b1 R 2 2 0 2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
120.38.2b2 R 2 2 0 2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
120.38.2c1 R 2 2 0 −1 0 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154
120.38.2c2 R 2 2 0 −1 0 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15
120.38.2c3 R 2 2 0 −1 0 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152
120.38.2c4 R 2 2 0 −1 0 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157
120.38.3a R 3 −1 1 0 −1 3 3 −1 −1 0 0 0 0
120.38.3b R 3 −1 −1 0 1 3 3 −1 −1 0 0 0 0
120.38.6a1 R 6 −2 0 0 0 3ζ5−2+3ζ52 3ζ5−1+3ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0
120.38.6a2 R 6 −2 0 0 0 3ζ5−1+3ζ5 3ζ5−2+3ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0

magma: CharacterTable(G);