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Properties

Label 30T34
Degree $30$
Order $120$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times S_4$

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Show commands: Magma

magma: G := TransitiveGroup(30, 34);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $34$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5\times S_4$
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:  (1,28,13)(2,27,14)(3,29,16)(4,30,15)(5,22,17)(6,21,18)(7,23,19)(8,24,20)(9,26,11)(10,25,12), (1,8,3,10,5,2,7,4,9,6)(11,22,13,23,16,26,17,28,19,29)(12,21,14,24,15,25,18,27,20,30)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$
$24$:  $S_4$
$30$:  $S_3 \times C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 5: $C_5$

Degree 6: $S_4$

Degree 10: None

Degree 15: $S_3 \times C_5$

Low degree siblings

20T34, 30T33, 40T64

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

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:  120.37
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 4A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 10B1 10B-1 10B3 10B-3 15A1 15A-1 15A2 15A-2 20A1 20A-1 20A3 20A-3
Size 1 3 6 8 6 1 1 1 1 3 3 3 3 6 6 6 6 8 8 8 8 6 6 6 6
2 P 1A 1A 1A 3A 2A 5A-1 5A-2 5A1 5A2 5A1 5A-1 5A-2 5A2 5A1 5A2 5A-1 5A-2 15A2 15A-1 15A1 15A-2 10A3 10A-3 10A1 10A-1
3 P 1A 2A 2B 1A 4A 5A1 5A2 5A-1 5A-2 10A3 10A-3 10A-1 10A1 10B-3 10B-1 10B3 10B1 5A1 5A2 5A-2 5A-1 20A-1 20A1 20A3 20A-3
5 P 1A 2A 2B 3A 4A 1A 1A 1A 1A 2A 2A 2A 2A 2B 2B 2B 2B 3A 3A 3A 3A 4A 4A 4A 4A
Type
120.37.1a R 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
120.37.1b R 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
120.37.1c1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1
120.37.1c2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5−1 ζ5
120.37.1c3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52
120.37.1c4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ52 ζ5−2
120.37.1d1 C 1 1 −1 1 −1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5 ζ5−1 ζ52 ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1
120.37.1d2 C 1 1 −1 1 −1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5
120.37.1d3 C 1 1 −1 1 −1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52
120.37.1d4 C 1 1 −1 1 −1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2
120.37.2a R 2 2 0 −1 0 2 2 2 2 2 2 2 2 0 0 0 0 −1 −1 −1 −1 0 0 0 0
120.37.2b1 C 2 2 0 −1 0 2ζ5−2 2ζ52 2ζ5 2ζ5−1 2ζ5−1 2ζ5 2ζ52 2ζ5−2 0 0 0 0 −ζ5 −ζ5−1 −ζ52 −ζ5−2 0 0 0 0
120.37.2b2 C 2 2 0 −1 0 2ζ52 2ζ5−2 2ζ5−1 2ζ5 2ζ5 2ζ5−1 2ζ5−2 2ζ52 0 0 0 0 −ζ5−1 −ζ5 −ζ5−2 −ζ52 0 0 0 0
120.37.2b3 C 2 2 0 −1 0 2ζ5−1 2ζ5 2ζ5−2 2ζ52 2ζ52 2ζ5−2 2ζ5 2ζ5−1 0 0 0 0 −ζ5−2 −ζ52 −ζ5 −ζ5−1 0 0 0 0
120.37.2b4 C 2 2 0 −1 0 2ζ5 2ζ5−1 2ζ52 2ζ5−2 2ζ5−2 2ζ52 2ζ5−1 2ζ5 0 0 0 0 −ζ52 −ζ5−2 −ζ5−1 −ζ5 0 0 0 0
120.37.3a R 3 −1 1 0 −1 3 3 3 3 −1 −1 −1 −1 1 1 1 1 0 0 0 0 −1 −1 −1 −1
120.37.3b R 3 −1 −1 0 1 3 3 3 3 −1 −1 −1 −1 −1 −1 −1 −1 0 0 0 0 1 1 1 1
120.37.3c1 C 3 −1 1 0 −1 3ζ5−2 3ζ52 3ζ5 3ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 0 0 0 0 −ζ52 −ζ5−2 −ζ5 −ζ5−1
120.37.3c2 C 3 −1 1 0 −1 3ζ52 3ζ5−2 3ζ5−1 3ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5−1 ζ5 ζ52 ζ5−2 0 0 0 0 −ζ5−2 −ζ52 −ζ5−1 −ζ5
120.37.3c3 C 3 −1 1 0 −1 3ζ5−1 3ζ5 3ζ5−2 3ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 0 0 0 0 −ζ5 −ζ5−1 −ζ5−2 −ζ52
120.37.3c4 C 3 −1 1 0 −1 3ζ5 3ζ5−1 3ζ52 3ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ52 ζ5−2 ζ5 ζ5−1 0 0 0 0 −ζ5−1 −ζ5 −ζ52 −ζ5−2
120.37.3d1 C 3 −1 −1 0 1 3ζ5−2 3ζ52 3ζ5 3ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−2 −ζ52 0 0 0 0 ζ52 ζ5−2 ζ5 ζ5−1
120.37.3d2 C 3 −1 −1 0 1 3ζ52 3ζ5−2 3ζ5−1 3ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ52 −ζ5−2 0 0 0 0 ζ5−2 ζ52 ζ5−1 ζ5
120.37.3d3 C 3 −1 −1 0 1 3ζ5−1 3ζ5 3ζ5−2 3ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ5−1 −ζ5 0 0 0 0 ζ5 ζ5−1 ζ5−2 ζ52
120.37.3d4 C 3 −1 −1 0 1 3ζ5 3ζ5−1 3ζ52 3ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5 −ζ5−1 0 0 0 0 ζ5−1 ζ5 ζ52 ζ5−2

magma: CharacterTable(G);