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Properties

Label 30T14
Degree $30$
Order $60$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{30}$

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Show commands: Magma

magma: G := TransitiveGroup(30, 14);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$10$:  $D_{5}$
$12$:  $D_{6}$
$20$:  $D_{10}$
$30$:  $D_{15}$

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: $D_{15}$

Low degree siblings

30T14

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 $2^{14},1^{2}$ $15$ $2$ $14$ $( 3,30)( 4,29)( 5,27)( 6,28)( 7,25)( 8,26)( 9,23)(10,24)(11,21)(12,22)(13,20)(14,19)(15,17)(16,18)$
2C $2^{15}$ $15$ $2$ $15$ $( 1,30)( 2,29)( 3,27)( 4,28)( 5,26)( 6,25)( 7,23)( 8,24)( 9,21)(10,22)(11,20)(12,19)(13,17)(14,18)(15,16)$
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 $5^{6}$ $2$ $5$ $24$ $( 1,14,25, 7,19)( 2,13,26, 8,20)( 3,16,28, 9,22)( 4,15,27,10,21)( 5,17,29,11,24)( 6,18,30,12,23)$
5A2 $5^{6}$ $2$ $5$ $24$ $( 1,25,19,14, 7)( 2,26,20,13, 8)( 3,28,22,16, 9)( 4,27,21,15,10)( 5,29,24,17,11)( 6,30,23,18,12)$
6A $6^{5}$ $2$ $6$ $25$ $( 1,22,11, 2,21,12)( 3,24,13, 4,23,14)( 5,26,15, 6,25,16)( 7,28,17, 8,27,18)( 9,29,20,10,30,19)$
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 $10^{3}$ $2$ $10$ $27$ $( 1,20, 7,26,14, 2,19, 8,25,13)( 3,21, 9,27,16, 4,22,10,28,15)( 5,23,11,30,17, 6,24,12,29,18)$
15A1 $15^{2}$ $2$ $15$ $28$ $( 1,29,27,25,24,21,19,17,15,14,11,10, 7, 5, 4)( 2,30,28,26,23,22,20,18,16,13,12, 9, 8, 6, 3)$
15A2 $15^{2}$ $2$ $15$ $28$ $( 1,24,15, 7,29,21,14, 5,27,19,11, 4,25,17,10)( 2,23,16, 8,30,22,13, 6,28,20,12, 3,26,18, 9)$
15A4 $15^{2}$ $2$ $15$ $28$ $( 1, 5,10,14,17,21,25,29, 4, 7,11,15,19,24,27)( 2, 6, 9,13,18,22,26,30, 3, 8,12,16,20,23,28)$
15A7 $15^{2}$ $2$ $15$ $28$ $( 1,17, 4,19, 5,21, 7,24,10,25,11,27,14,29,15)( 2,18, 3,20, 6,22, 8,23, 9,26,12,28,13,30,16)$
30A1 $30$ $2$ $30$ $29$ $( 1,16,29,13,27,12,25, 9,24, 8,21, 6,19, 3,17, 2,15,30,14,28,11,26,10,23, 7,22, 5,20, 4,18)$
30A7 $30$ $2$ $30$ $29$ $( 1, 3, 5, 8,10,12,14,16,17,20,21,23,25,28,29, 2, 4, 6, 7, 9,11,13,15,18,19,22,24,26,27,30)$
30A11 $30$ $2$ $30$ $29$ $( 1,28,24,20,15,12, 7, 3,29,26,21,18,14, 9, 5, 2,27,23,19,16,11, 8, 4,30,25,22,17,13,10, 6)$
30A13 $30$ $2$ $30$ $29$ $( 1, 9,17,26, 4,12,19,28, 5,13,21,30, 7,16,24, 2,10,18,25, 3,11,20,27, 6,14,22,29, 8,15,23)$

Malle's constant $a(G)$:     $1/14$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  60.12
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 5A1 5A2 6A 10A1 10A3 15A1 15A2 15A4 15A7 30A1 30A7 30A11 30A13
Size 1 1 15 15 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A 5A2 5A1 3A 5A1 5A2 15A1 15A4 15A2 15A7 15A7 15A1 15A2 15A4
3 P 1A 2A 2B 2C 1A 5A2 5A1 2A 10A3 10A1 5A2 5A2 5A1 5A1 10A3 10A1 10A3 10A1
5 P 1A 2A 2B 2C 3A 1A 1A 6A 2A 2A 3A 3A 3A 3A 6A 6A 6A 6A
Type
60.12.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
60.12.1b R 1 −1 −1 1 1 1 1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
60.12.1c R 1 −1 1 −1 1 1 1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
60.12.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
60.12.2a R 2 2 0 0 −1 2 2 −1 2 2 −1 −1 −1 −1 −1 −1 −1 −1
60.12.2b R 2 −2 0 0 −1 2 2 1 −2 −2 −1 −1 −1 −1 1 1 1 1
60.12.2c1 R 2 2 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
60.12.2c2 R 2 2 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
60.12.2d1 R 2 −2 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 −2 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
60.12.2d2 R 2 −2 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 −2 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
60.12.2e1 R 2 2 0 0 −1 ζ15−6+ζ156 ζ15−3+ζ153 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−4+ζ154 ζ15−2+ζ152 ζ15−1+ζ15 ζ15−7+ζ157
60.12.2e2 R 2 2 0 0 −1 ζ15−6+ζ156 ζ15−3+ζ153 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−1+ζ15 ζ15−7+ζ157 ζ15−4+ζ154 ζ15−2+ζ152
60.12.2e3 R 2 2 0 0 −1 ζ15−3+ζ153 ζ15−6+ζ156 −1 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−2+ζ152 ζ15−1+ζ15 ζ15−7+ζ157 ζ15−4+ζ154
60.12.2e4 R 2 2 0 0 −1 ζ15−3+ζ153 ζ15−6+ζ156 −1 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−7+ζ157 ζ15−4+ζ154 ζ15−2+ζ152 ζ15−1+ζ15
60.12.2f1 R 2 −2 0 0 −1 ζ15−6+ζ156 ζ15−3+ζ153 1 −ζ15−3−ζ153 −ζ15−6−ζ156 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 −ζ15−4−ζ154 −ζ15−2−ζ152 −ζ15−1−ζ15 −ζ15−7−ζ157
60.12.2f2 R 2 −2 0 0 −1 ζ15−6+ζ156 ζ15−3+ζ153 1 −ζ15−3−ζ153 −ζ15−6−ζ156 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 −ζ15−1−ζ15 −ζ15−7−ζ157 −ζ15−4−ζ154 −ζ15−2−ζ152
60.12.2f3 R 2 −2 0 0 −1 ζ15−3+ζ153 ζ15−6+ζ156 1 −ζ15−6−ζ156 −ζ15−3−ζ153 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 −ζ15−2−ζ152 −ζ15−1−ζ15 −ζ15−7−ζ157 −ζ15−4−ζ154
60.12.2f4 R 2 −2 0 0 −1 ζ15−3+ζ153 ζ15−6+ζ156 1 −ζ15−6−ζ156 −ζ15−3−ζ153 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 −ζ15−7−ζ157 −ζ15−4−ζ154 −ζ15−2−ζ152 −ζ15−1−ζ15

magma: CharacterTable(G);