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Properties

Label 30T24
30T24 1 4 1->4 7 1->7 2 3 2->3 8 2->8 17 3->17 29 3->29 18 4->18 30 4->30 5 5->2 22 5->22 6 6->1 21 6->21 13 7->13 16 7->16 14 8->14 15 8->15 9 9->5 9->30 10 10->6 10->29 11 11->14 28 11->28 12 12->13 27 12->27 20 13->20 13->27 19 14->19 14->28 15->11 15->12 16->11 16->12 17->4 26 17->26 18->3 25 18->25 19->9 19->25 20->10 20->26 21->17 24 21->24 22->18 23 22->23 23->7 23->10 24->8 24->9 25->2 25->22 26->1 26->21 27->5 27->23 28->6 28->24 29->15 29->19 30->16 30->20
Degree $30$
Order $120$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times F_5$

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Copy content magma:G := TransitiveGroup(30, 24);
 

Group invariants

Abstract group:  $S_3\times F_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $120=2^{3} \cdot 3 \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $30$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $24$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,7,13,20,26)(2,8,14,19,25)(3,29,15,11,28,24,9,5,22,18)(4,30,16,12,27,23,10,6,21,17)$, $(1,4,18,25,22,23,7,16,11,14,28,6)(2,3,17,26,21,24,8,15,12,13,27,5)(9,30,20,10,29,19)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$20$:  $F_5$
$24$:  $S_3 \times C_4$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $F_5$

Degree 6: $D_{6}$

Degree 10: $F_5$

Degree 15: $F_5 \times S_3$

Low degree siblings

15T11, 30T23, 30T32

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

Malle's constant $a(G)$:     $1/10$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 5A 6A 10A 12A1 12A-1 15A
Size 1 3 5 15 2 5 5 15 15 4 10 12 10 10 8
2 P 1A 1A 1A 1A 3A 2B 2B 2B 2B 5A 3A 5A 6A 6A 15A
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 5A 2B 10A 4A1 4A-1 5A
5 P 1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 1A 6A 2A 12A1 12A-1 3A
Type
120.36.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
120.36.1b R 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1
120.36.1c R 1 −1 1 −1 1 1 1 −1 −1 1 1 −1 1 1 1
120.36.1d R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1
120.36.1e1 C 1 −1 −1 1 1 −i i i −i 1 −1 −1 i −i 1
120.36.1e2 C 1 −1 −1 1 1 i −i −i i 1 −1 −1 −i i 1
120.36.1f1 C 1 1 −1 −1 1 −i i −i i 1 −1 1 i −i 1
120.36.1f2 C 1 1 −1 −1 1 i −i i −i 1 −1 1 −i i 1
120.36.2a R 2 0 2 0 −1 2 2 0 0 2 −1 0 −1 −1 −1
120.36.2b R 2 0 2 0 −1 −2 −2 0 0 2 −1 0 1 1 −1
120.36.2c1 C 2 0 −2 0 −1 −2i 2i 0 0 2 1 0 −i i −1
120.36.2c2 C 2 0 −2 0 −1 2i −2i 0 0 2 1 0 i −i −1
120.36.4a R 4 4 0 0 4 0 0 0 0 −1 0 −1 0 0 −1
120.36.4b R 4 −4 0 0 4 0 0 0 0 −1 0 1 0 0 −1
120.36.8a R 8 0 0 0 −4 0 0 0 0 −2 0 0 0 0 1

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Regular extensions

Data not computed