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Properties

Label 30T33
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, 33);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
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:  (1,29,6,23,9,28,3,21,8,26)(2,30,5,24,10,27,4,22,7,25)(11,14,15,17,19)(12,13,16,18,20), (1,27,13)(2,28,14)(3,30,16)(4,29,15)(5,21,17)(6,22,18)(7,23,19)(8,24,20)(9,25,12)(10,26,11)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$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, 30T34, 40T64

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{30}$ $1$ $1$ $0$ $()$
2A $2^{10},1^{10}$ $3$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(21,22)(23,24)(25,26)(27,28)(29,30)$
2B $2^{10},1^{10}$ $6$ $2$ $10$ $(11,25)(12,26)(13,28)(14,27)(15,30)(16,29)(17,22)(18,21)(19,24)(20,23)$
3A $3^{10}$ $8$ $3$ $20$ $( 1,28,13)( 2,27,14)( 3,29,16)( 4,30,15)( 5,22,17)( 6,21,18)( 7,24,19)( 8,23,20)( 9,26,12)(10,25,11)$
4A $4^{5},2^{5}$ $6$ $4$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,26,12,25)(13,27,14,28)(15,29,16,30)(17,21,18,22)(19,23,20,24)$
5A1 $5^{6}$ $1$ $5$ $24$ $( 1, 8, 3, 9, 6)( 2, 7, 4,10, 5)(11,17,14,19,15)(12,18,13,20,16)(21,28,23,29,26)(22,27,24,30,25)$
5A-1 $5^{6}$ $1$ $5$ $24$ $( 1, 6, 9, 3, 8)( 2, 5,10, 4, 7)(11,15,19,14,17)(12,16,20,13,18)(21,26,29,23,28)(22,25,30,24,27)$
5A2 $5^{6}$ $1$ $5$ $24$ $( 1, 3, 6, 8, 9)( 2, 4, 5, 7,10)(11,14,15,17,19)(12,13,16,18,20)(21,23,26,28,29)(22,24,25,27,30)$
5A-2 $5^{6}$ $1$ $5$ $24$ $( 1, 9, 8, 6, 3)( 2,10, 7, 5, 4)(11,19,17,15,14)(12,20,18,16,13)(21,29,28,26,23)(22,30,27,25,24)$
10A1 $10^{2},5^{2}$ $3$ $10$ $26$ $( 1, 5, 9, 4, 8, 2, 6,10, 3, 7)(11,15,19,14,17)(12,16,20,13,18)(21,25,29,24,28,22,26,30,23,27)$
10A-1 $10^{2},5^{2}$ $3$ $10$ $26$ $( 1, 7, 3,10, 6, 2, 8, 4, 9, 5)(11,17,14,19,15)(12,18,13,20,16)(21,27,23,30,26,22,28,24,29,25)$
10A3 $10^{2},5^{2}$ $3$ $10$ $26$ $( 1, 4, 6, 7, 9, 2, 3, 5, 8,10)(11,14,15,17,19)(12,13,16,18,20)(21,24,26,27,29,22,23,25,28,30)$
10A-3 $10^{2},5^{2}$ $3$ $10$ $26$ $( 1,10, 8, 5, 3, 2, 9, 7, 6, 4)(11,19,17,15,14)(12,20,18,16,13)(21,30,28,25,23,22,29,27,26,24)$
10B1 $10^{2},5^{2}$ $6$ $10$ $26$ $( 1, 3, 6, 8, 9)( 2, 4, 5, 7,10)(11,27,15,22,19,25,14,30,17,24)(12,28,16,21,20,26,13,29,18,23)$
10B-1 $10^{2},5^{2}$ $6$ $10$ $26$ $( 1, 9, 8, 6, 3)( 2,10, 7, 5, 4)(11,24,17,30,14,25,19,22,15,27)(12,23,18,29,13,26,20,21,16,28)$
10B3 $10^{2},5^{2}$ $6$ $10$ $26$ $( 1, 8, 3, 9, 6)( 2, 7, 4,10, 5)(11,22,14,24,15,25,17,27,19,30)(12,21,13,23,16,26,18,28,20,29)$
10B-3 $10^{2},5^{2}$ $6$ $10$ $26$ $( 1, 6, 9, 3, 8)( 2, 5,10, 4, 7)(11,30,19,27,17,25,15,24,14,22)(12,29,20,28,18,26,16,23,13,21)$
15A1 $15^{2}$ $8$ $15$ $28$ $( 1,23,16, 9,21,13, 8,29,12, 6,28,20, 3,26,18)( 2,24,15,10,22,14, 7,30,11, 5,27,19, 4,25,17)$
15A-1 $15^{2}$ $8$ $15$ $28$ $( 1,29,18, 8,26,13, 3,21,20, 9,28,16, 6,23,12)( 2,30,17, 7,25,14, 4,22,19,10,27,15, 5,24,11)$
15A2 $15^{2}$ $8$ $15$ $28$ $( 1,21,12, 3,23,13, 6,26,16, 8,28,18, 9,29,20)( 2,22,11, 4,24,14, 5,25,15, 7,27,17,10,30,19)$
15A-2 $15^{2}$ $8$ $15$ $28$ $( 1,26,20, 6,29,13, 9,23,18, 3,28,12, 8,21,16)( 2,25,19, 5,30,14,10,24,17, 4,27,11, 7,22,15)$
20A1 $20,10$ $6$ $20$ $28$ $( 1,10, 8, 5, 3, 2, 9, 7, 6, 4)(11,23,18,30,14,26,20,22,15,28,12,24,17,29,13,25,19,21,16,27)$
20A-1 $20,10$ $6$ $20$ $28$ $( 1, 4, 6, 7, 9, 2, 3, 5, 8,10)(11,28,16,22,19,26,13,30,17,23,12,27,15,21,20,25,14,29,18,24)$
20A3 $20,10$ $6$ $20$ $28$ $( 1, 5, 9, 4, 8, 2, 6,10, 3, 7)(11,29,20,27,17,26,16,24,14,21,12,30,19,28,18,25,15,23,13,22)$
20A-3 $20,10$ $6$ $20$ $28$ $( 1, 7, 3,10, 6, 2, 8, 4, 9, 5)(11,21,13,24,15,26,18,27,19,29,12,22,14,23,16,25,17,28,20,30)$

Malle's constant $a(G)$:     $1/10$

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 5A2 5A-2 5A-1 5A1 5A-2 5A2 5A-1 5A1 5A-1 5A1 5A2 5A-2 15A-1 15A-2 15A1 15A2 10A1 10A-1 10A3 10A-3
3 P 1A 2A 2B 1A 4A 5A-2 5A2 5A1 5A-1 10A-3 10A3 10A1 10A-1 10B-3 10B3 10B1 10B-1 5A-2 5A1 5A2 5A-1 20A3 20A-3 20A-1 20A1
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);