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Properties

Label 30T21
30T21 1 6 1->6 11 1->11 14 1->14 2 5 2->5 12 2->12 13 2->13 3 4 3->4 3->12 4->11 10 5->10 25 5->25 9 6->9 26 6->26 7 17 7->17 30 7->30 8 18 8->18 29 8->29 27 9->27 28 10->28 11->26 12->25 23 13->23 24 13->24 14->23 14->24 15 22 15->22 15->29 16 21 16->21 16->30 20 17->20 17->27 19 18->19 18->28 19->25 19->29 20->26 20->30 21->24 22->23
Degree $30$
Order $120$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_{10}$

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Copy content magma:G := TransitiveGroup(30, 21);
 

Group invariants

Abstract group:  $S_3\times D_{10}$
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 $t$:  $21$
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:  $(1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(15,29)(16,30)(17,27)(18,28)(19,25)(20,26)(21,24)(22,23)$, $(1,6)(2,5)(3,4)(7,30)(8,29)(9,27)(10,28)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)$, $(1,11)(2,12)(5,25)(6,26)(7,17)(8,18)(13,23)(14,24)(19,29)(20,30)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$
$8$:  $C_2^3$
$10$:  $D_{5}$
$12$:  $D_{6}$ x 3
$20$:  $D_{10}$ x 3
$24$:  $S_3 \times C_2^2$
$40$:  20T8
$60$:  $D_5\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 6: $D_{6}$

Degree 10: $D_{10}$

Degree 15: $D_5\times S_3$

Low degree siblings

30T21 x 3

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 $2^{15}$ $1$ $2$ $15$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)$
2B $2^{10},1^{10}$ $3$ $2$ $10$ $( 1,11)( 2,12)( 5,25)( 6,26)( 7,17)( 8,18)(13,23)(14,24)(19,29)(20,30)$
2C $2^{15}$ $3$ $2$ $15$ $( 1,22)( 2,21)( 3,14)( 4,13)( 5, 6)( 7,28)( 8,27)( 9,19)(10,20)(11,12)(15,26)(16,25)(17,18)(23,24)(29,30)$
2D $2^{15}$ $5$ $2$ $15$ $( 1,20)( 2,19)( 3,27)( 4,28)( 5, 6)( 7,13)( 8,14)( 9,21)(10,22)(11,30)(12,29)(15,16)(17,23)(18,24)(25,26)$
2E $2^{12},1^{6}$ $5$ $2$ $12$ $( 1, 7)( 2, 8)( 3,16)( 4,15)( 5,24)( 6,23)(11,17)(12,18)(13,26)(14,25)(21,27)(22,28)$
2F $2^{15}$ $15$ $2$ $15$ $( 1,30)( 2,29)( 3,27)( 4,28)( 5,26)( 6,25)( 7,23)( 8,24)( 9,21)(10,22)(11,20)(12,19)(13,17)(14,18)(15,16)$
2G $2^{14},1^{2}$ $15$ $2$ $14$ $( 1,14)( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)(15,29)(16,30)(17,27)(18,28)(19,25)(20,26)(21,24)(22,23)$
3A $3^{10}$ $2$ $3$ $20$ $( 1,11,21)( 2,12,22)( 3,13,23)( 4,14,24)( 5,15,25)( 6,16,26)( 7,17,27)( 8,18,28)( 9,20,30)(10,19,29)$
5A1 $5^{6}$ $2$ $5$ $24$ $( 1,25,19,14, 7)( 2,26,20,13, 8)( 3,28,22,16, 9)( 4,27,21,15,10)( 5,29,24,17,11)( 6,30,23,18,12)$
5A2 $5^{6}$ $2$ $5$ $24$ $( 1,19, 7,25,14)( 2,20, 8,26,13)( 3,22, 9,28,16)( 4,21,10,27,15)( 5,24,11,29,17)( 6,23,12,30,18)$
6A $6^{5}$ $2$ $6$ $25$ $( 1,22,11, 2,21,12)( 3,24,13, 4,23,14)( 5,26,15, 6,25,16)( 7,28,17, 8,27,18)( 9,29,20,10,30,19)$
6B $6^{5}$ $10$ $6$ $25$ $( 1, 9,11,20,21,30)( 2,10,12,19,22,29)( 3,17,13,27,23, 7)( 4,18,14,28,24, 8)( 5,26,15, 6,25,16)$
6C $6^{4},3^{2}$ $10$ $6$ $24$ $( 1,17,21, 7,11,27)( 2,18,22, 8,12,28)( 3,26,23,16,13, 6)( 4,25,24,15,14, 5)( 9,20,30)(10,19,29)$
10A1 $10^{3}$ $2$ $10$ $27$ $( 1, 8,14,20,25, 2, 7,13,19,26)( 3,10,16,21,28, 4, 9,15,22,27)( 5,12,17,23,29, 6,11,18,24,30)$
10A3 $10^{3}$ $2$ $10$ $27$ $( 1,20, 7,26,14, 2,19, 8,25,13)( 3,21, 9,27,16, 4,22,10,28,15)( 5,23,11,30,17, 6,24,12,29,18)$
10B1 $10^{2},5^{2}$ $6$ $10$ $26$ $( 1,24,25,17,19,11,14, 5, 7,29)( 2,23,26,18,20,12,13, 6, 8,30)( 3,16,28, 9,22)( 4,15,27,10,21)$
10B3 $10^{2},5^{2}$ $6$ $10$ $26$ $( 1,25,19,14, 7)( 2,26,20,13, 8)( 3,18,22, 6, 9,23,28,12,16,30)( 4,17,21, 5,10,24,27,11,15,29)$
10C1 $10^{3}$ $6$ $10$ $27$ $( 1, 3,25,28,19,22,14,16, 7, 9)( 2, 4,26,27,20,21,13,15, 8,10)( 5,18,29,12,24, 6,17,30,11,23)$
10C3 $10^{3}$ $6$ $10$ $27$ $( 1,16,19, 3, 7,22,25, 9,14,28)( 2,15,20, 4, 8,21,26,10,13,27)( 5,30,24,18,11, 6,29,23,17,12)$
15A1 $15^{2}$ $4$ $15$ $28$ $( 1,15,29,14,27,11,25,10,24, 7,21, 5,19, 4,17)( 2,16,30,13,28,12,26, 9,23, 8,22, 6,20, 3,18)$
15A2 $15^{2}$ $4$ $15$ $28$ $( 1,29,27,25,24,21,19,17,15,14,11,10, 7, 5, 4)( 2,30,28,26,23,22,20,18,16,13,12, 9, 8, 6, 3)$
30A1 $30$ $4$ $30$ $29$ $( 1,23,15, 8,29,22,14, 6,27,20,11, 3,25,18,10, 2,24,16, 7,30,21,13, 5,28,19,12, 4,26,17, 9)$
30A7 $30$ $4$ $30$ $29$ $( 1, 6,10,13,17,22,25,30, 4, 8,11,16,19,23,27, 2, 5, 9,14,18,21,26,29, 3, 7,12,15,20,24,28)$

Malle's constant $a(G)$:     $1/10$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 3A 5A1 5A2 6A 6B 6C 10A1 10A3 10B1 10B3 10C1 10C3 15A1 15A2 30A1 30A7
Size 1 1 3 3 5 5 15 15 2 2 2 2 10 10 2 2 6 6 6 6 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 5A2 5A1 3A 3A 3A 5A2 5A1 5A1 5A2 5A1 5A2 15A2 15A1 15A1 15A2
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 5A2 5A1 2A 2D 2E 10A3 10A1 10B3 10B1 10C3 10C1 5A2 5A1 10A1 10A3
5 P 1A 2A 2B 2C 2D 2E 2F 2G 3A 1A 1A 6A 6B 6C 2A 2A 2B 2B 2C 2C 3A 3A 6A 6A
Type
120.42.1a 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
120.42.1b 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
120.42.1c 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
120.42.1d 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
120.42.1e 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
120.42.1f 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
120.42.1g 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
120.42.1h 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
120.42.2a R 2 2 0 0 2 2 0 0 −1 2 2 −1 −1 −1 2 2 0 0 0 0 −1 −1 −1 −1
120.42.2b R 2 −2 0 0 −2 2 0 0 −1 2 2 1 1 −1 −2 −2 0 0 0 0 −1 −1 1 1
120.42.2c R 2 −2 0 0 2 −2 0 0 −1 2 2 1 −1 1 −2 −2 0 0 0 0 −1 −1 1 1
120.42.2d R 2 2 0 0 −2 −2 0 0 −1 2 2 −1 1 1 2 2 0 0 0 0 −1 −1 −1 −1
120.42.2e1 R 2 2 2 2 0 0 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
120.42.2e2 R 2 2 2 2 0 0 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
120.42.2f1 R 2 −2 −2 2 0 0 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 −2 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
120.42.2f2 R 2 −2 −2 2 0 0 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 −2 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
120.42.2g1 R 2 −2 2 −2 0 0 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 −2 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
120.42.2g2 R 2 −2 2 −2 0 0 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 −2 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
120.42.2h1 R 2 2 −2 −2 0 0 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
120.42.2h2 R 2 2 −2 −2 0 0 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
120.42.4a1 R 4 4 0 0 0 0 0 0 −2 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −2 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
120.42.4a2 R 4 4 0 0 0 0 0 0 −2 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −2 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
120.42.4b1 R 4 −4 0 0 0 0 0 0 −2 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2 0 0 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52
120.42.4b2 R 4 −4 0 0 0 0 0 0 −2 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2 0 0 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed