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Properties

Label 30T1
Degree $30$
Order $30$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_{30}$

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Show commands: Magma

magma: G := TransitiveGroup(30, 1);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$5$:  $C_5$
$6$:  $C_6$
$10$:  $C_{10}$
$15$:  $C_{15}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 5: $C_5$

Degree 6: $C_6$

Degree 10: $C_{10}$

Degree 15: $C_{15}$

Low degree siblings

There are no siblings 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, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)$
$ 15, 15 $ $1$ $15$ $( 1, 3, 5, 7, 9,12,13,15,18,20,21,24,26,27,30)( 2, 4, 6, 8,10,11,14,16,17,19, 22,23,25,28,29)$
$ 30 $ $1$ $30$ $( 1, 4, 5, 8, 9,11,13,16,18,19,21,23,26,28,30, 2, 3, 6, 7,10,12,14,15,17,20, 22,24,25,27,29)$
$ 15, 15 $ $1$ $15$ $( 1, 5, 9,13,18,21,26,30, 3, 7,12,15,20,24,27)( 2, 6,10,14,17,22,25,29, 4, 8, 11,16,19,23,28)$
$ 30 $ $1$ $30$ $( 1, 6, 9,14,18,22,26,29, 3, 8,12,16,20,23,27, 2, 5,10,13,17,21,25,30, 4, 7, 11,15,19,24,28)$
$ 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 7,13,20,26)( 2, 8,14,19,25)( 3, 9,15,21,27)( 4,10,16,22,28) ( 5,12,18,24,30)( 6,11,17,23,29)$
$ 10, 10, 10 $ $1$ $10$ $( 1, 8,13,19,26, 2, 7,14,20,25)( 3,10,15,22,27, 4, 9,16,21,28)( 5,11,18,23,30, 6,12,17,24,29)$
$ 15, 15 $ $1$ $15$ $( 1, 9,18,26, 3,12,20,27, 5,13,21,30, 7,15,24)( 2,10,17,25, 4,11,19,28, 6,14, 22,29, 8,16,23)$
$ 30 $ $1$ $30$ $( 1,10,18,25, 3,11,20,28, 5,14,21,29, 7,16,24, 2, 9,17,26, 4,12,19,27, 6,13, 22,30, 8,15,23)$
$ 6, 6, 6, 6, 6 $ $1$ $6$ $( 1,11,21, 2,12,22)( 3,14,24, 4,13,23)( 5,16,26, 6,15,25)( 7,17,27, 8,18,28) ( 9,19,30,10,20,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,12,21)( 2,11,22)( 3,13,24)( 4,14,23)( 5,15,26)( 6,16,25)( 7,18,27) ( 8,17,28)( 9,20,30)(10,19,29)$
$ 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,13,26, 7,20)( 2,14,25, 8,19)( 3,15,27, 9,21)( 4,16,28,10,22) ( 5,18,30,12,24)( 6,17,29,11,23)$
$ 10, 10, 10 $ $1$ $10$ $( 1,14,26, 8,20, 2,13,25, 7,19)( 3,16,27,10,21, 4,15,28, 9,22)( 5,17,30,11,24, 6,18,29,12,23)$
$ 15, 15 $ $1$ $15$ $( 1,15,30,13,27,12,26, 9,24, 7,21, 5,20, 3,18)( 2,16,29,14,28,11,25,10,23, 8, 22, 6,19, 4,17)$
$ 30 $ $1$ $30$ $( 1,16,30,14,27,11,26,10,24, 8,21, 6,20, 4,18, 2,15,29,13,28,12,25, 9,23, 7, 22, 5,19, 3,17)$
$ 30 $ $1$ $30$ $( 1,17, 3,19, 5,22, 7,23, 9,25,12,28,13,29,15, 2,18, 4,20, 6,21, 8,24,10,26, 11,27,14,30,16)$
$ 15, 15 $ $1$ $15$ $( 1,18, 3,20, 5,21, 7,24, 9,26,12,27,13,30,15)( 2,17, 4,19, 6,22, 8,23,10,25, 11,28,14,29,16)$
$ 10, 10, 10 $ $1$ $10$ $( 1,19, 7,25,13, 2,20, 8,26,14)( 3,22, 9,28,15, 4,21,10,27,16)( 5,23,12,29,18, 6,24,11,30,17)$
$ 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,20, 7,26,13)( 2,19, 8,25,14)( 3,21, 9,27,15)( 4,22,10,28,16) ( 5,24,12,30,18)( 6,23,11,29,17)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,21,12)( 2,22,11)( 3,24,13)( 4,23,14)( 5,26,15)( 6,25,16)( 7,27,18) ( 8,28,17)( 9,30,20)(10,29,19)$
$ 6, 6, 6, 6, 6 $ $1$ $6$ $( 1,22,12, 2,21,11)( 3,23,13, 4,24,14)( 5,25,15, 6,26,16)( 7,28,18, 8,27,17) ( 9,29,20,10,30,19)$
$ 30 $ $1$ $30$ $( 1,23,15, 8,30,22,13, 6,27,19,12, 4,26,17, 9, 2,24,16, 7,29,21,14, 5,28,20, 11, 3,25,18,10)$
$ 15, 15 $ $1$ $15$ $( 1,24,15, 7,30,21,13, 5,27,20,12, 3,26,18, 9)( 2,23,16, 8,29,22,14, 6,28,19, 11, 4,25,17,10)$
$ 10, 10, 10 $ $1$ $10$ $( 1,25,20,14, 7, 2,26,19,13, 8)( 3,28,21,16, 9, 4,27,22,15,10)( 5,29,24,17,12, 6,30,23,18,11)$
$ 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,26,20,13, 7)( 2,25,19,14, 8)( 3,27,21,15, 9)( 4,28,22,16,10) ( 5,30,24,18,12)( 6,29,23,17,11)$
$ 15, 15 $ $1$ $15$ $( 1,27,24,20,15,12, 7, 3,30,26,21,18,13, 9, 5)( 2,28,23,19,16,11, 8, 4,29,25, 22,17,14,10, 6)$
$ 30 $ $1$ $30$ $( 1,28,24,19,15,11, 7, 4,30,25,21,17,13,10, 5, 2,27,23,20,16,12, 8, 3,29,26, 22,18,14, 9, 6)$
$ 30 $ $1$ $30$ $( 1,29,27,25,24,22,20,17,15,14,12,10, 7, 6, 3, 2,30,28,26,23,21,19,18,16,13, 11, 9, 8, 5, 4)$
$ 15, 15 $ $1$ $15$ $( 1,30,27,26,24,21,20,18,15,13,12, 9, 7, 5, 3)( 2,29,28,25,23,22,19,17,16,14, 11,10, 8, 6, 4)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $30=2 \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  30.4
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 5A1 5A-1 5A2 5A-2 6A1 6A-1 10A1 10A-1 10A3 10A-3 15A1 15A-1 15A2 15A-2 15A4 15A-4 15A7 15A-7 30A1 30A-1 30A7 30A-7 30A11 30A-11 30A13 30A-13
Size 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 P 1A 1A 3A-1 3A1 5A2 5A-2 5A-1 5A1 3A1 3A-1 5A1 5A-1 5A-2 5A2 15A-1 15A4 15A1 15A2 15A-7 15A-4 15A-2 15A7 15A-1 15A-7 15A2 15A-2 15A4 15A7 15A1 15A-4
3 P 1A 2A 1A 1A 5A-2 5A2 5A1 5A-1 2A 2A 10A3 10A-3 10A-1 10A1 5A2 5A2 5A-2 5A1 5A-1 5A-2 5A-1 5A1 10A-1 10A3 10A-3 10A3 10A-1 10A-3 10A1 10A1
5 P 1A 2A 3A-1 3A1 1A 1A 1A 1A 6A-1 6A1 2A 2A 2A 2A 3A1 3A-1 3A-1 3A1 3A1 3A1 3A-1 3A-1 6A-1 6A-1 6A-1 6A1 6A1 6A1 6A1 6A-1
Type
30.4.1a 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 1 1 1 1 1
30.4.1b 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 −1 −1 −1 −1 −1
30.4.1c1 C 1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
30.4.1c2 C 1 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
30.4.1d1 C 1 −1 ζ3−1 ζ3 1 1 1 1 −ζ3 −ζ3−1 −1 −1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
30.4.1d2 C 1 −1 ζ3 ζ3−1 1 1 1 1 −ζ3−1 −ζ3 −1 −1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
30.4.1e1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5
30.4.1e2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1
30.4.1e3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2
30.4.1e4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52
30.4.1f1 C 1 −1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 −1 −1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ5−1 −ζ5
30.4.1f2 C 1 −1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 −1 −1 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5 −ζ5−1
30.4.1f3 C 1 −1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 −1 −1 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ52 −ζ5−2
30.4.1f4 C 1 −1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 −1 −1 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−2 −ζ52
30.4.1g1 C 1 1 ζ15−5 ζ155 ζ15−6 ζ156 ζ153 ζ15−3 ζ155 ζ15−5 ζ15−3 ζ153 ζ156 ζ15−6 ζ15−7 ζ157 ζ15 ζ15−1 ζ152 ζ15−2 ζ15−4 ζ154 ζ154 ζ15−4 ζ15−2 ζ152 ζ15−1 ζ15 ζ157 ζ15−7
30.4.1g2 C 1 1 ζ155 ζ15−5 ζ156 ζ15−6 ζ15−3 ζ153 ζ15−5 ζ155 ζ153 ζ15−3 ζ15−6 ζ156 ζ157 ζ15−7 ζ15−1 ζ15 ζ15−2 ζ152 ζ154 ζ15−4 ζ15−4 ζ154 ζ152 ζ15−2 ζ15 ζ15−1 ζ15−7 ζ157
30.4.1g3 C 1 1 ζ15−5 ζ155 ζ156 ζ15−6 ζ15−3 ζ153 ζ155 ζ15−5 ζ153 ζ15−3 ζ15−6 ζ156 ζ152 ζ15−2 ζ154 ζ15−4 ζ15−7 ζ157 ζ15−1 ζ15 ζ15 ζ15−1 ζ157 ζ15−7 ζ15−4 ζ154 ζ15−2 ζ152
30.4.1g4 C 1 1 ζ155 ζ15−5 ζ15−6 ζ156 ζ153 ζ15−3 ζ15−5 ζ155 ζ15−3 ζ153 ζ156 ζ15−6 ζ15−2 ζ152 ζ15−4 ζ154 ζ157 ζ15−7 ζ15 ζ15−1 ζ15−1 ζ15 ζ15−7 ζ157 ζ154 ζ15−4 ζ152 ζ15−2
30.4.1g5 C 1 1 ζ15−5 ζ155 ζ15−3 ζ153 ζ15−6 ζ156 ζ155 ζ15−5 ζ156 ζ15−6 ζ153 ζ15−3 ζ15−1 ζ15 ζ15−2 ζ152 ζ15−4 ζ154 ζ15−7 ζ157 ζ157 ζ15−7 ζ154 ζ15−4 ζ152 ζ15−2 ζ15 ζ15−1
30.4.1g6 C 1 1 ζ155 ζ15−5 ζ153 ζ15−3 ζ156 ζ15−6 ζ15−5 ζ155 ζ15−6 ζ156 ζ15−3 ζ153 ζ15 ζ15−1 ζ152 ζ15−2 ζ154 ζ15−4 ζ157 ζ15−7 ζ15−7 ζ157 ζ15−4 ζ154 ζ15−2 ζ152 ζ15−1 ζ15
30.4.1g7 C 1 1 ζ15−5 ζ155 ζ153 ζ15−3 ζ156 ζ15−6 ζ155 ζ15−5 ζ15−6 ζ156 ζ15−3 ζ153 ζ15−4 ζ154 ζ157 ζ15−7 ζ15−1 ζ15 ζ152 ζ15−2 ζ15−2 ζ152 ζ15 ζ15−1 ζ15−7 ζ157 ζ154 ζ15−4
30.4.1g8 C 1 1 ζ155 ζ15−5 ζ15−3 ζ153 ζ15−6 ζ156 ζ15−5 ζ155 ζ156 ζ15−6 ζ153 ζ15−3 ζ154 ζ15−4 ζ15−7 ζ157 ζ15 ζ15−1 ζ15−2 ζ152 ζ152 ζ15−2 ζ15−1 ζ15 ζ157 ζ15−7 ζ15−4 ζ154
30.4.1h1 C 1 −1 ζ15−5 ζ155 ζ15−6 ζ156 ζ153 ζ15−3 −ζ155 −ζ15−5 −ζ15−3 −ζ153 −ζ156 −ζ15−6 ζ15−7 ζ157 ζ15 ζ15−1 ζ152 ζ15−2 ζ15−4 ζ154 −ζ154 −ζ15−4 −ζ15−2 −ζ152 −ζ15−1 −ζ15 −ζ157 −ζ15−7
30.4.1h2 C 1 −1 ζ155 ζ15−5 ζ156 ζ15−6 ζ15−3 ζ153 −ζ15−5 −ζ155 −ζ153 −ζ15−3 −ζ15−6 −ζ156 ζ157 ζ15−7 ζ15−1 ζ15 ζ15−2 ζ152 ζ154 ζ15−4 −ζ15−4 −ζ154 −ζ152 −ζ15−2 −ζ15 −ζ15−1 −ζ15−7 −ζ157
30.4.1h3 C 1 −1 ζ15−5 ζ155 ζ156 ζ15−6 ζ15−3 ζ153 −ζ155 −ζ15−5 −ζ153 −ζ15−3 −ζ15−6 −ζ156 ζ152 ζ15−2 ζ154 ζ15−4 ζ15−7 ζ157 ζ15−1 ζ15 −ζ15 −ζ15−1 −ζ157 −ζ15−7 −ζ15−4 −ζ154 −ζ15−2 −ζ152
30.4.1h4 C 1 −1 ζ155 ζ15−5 ζ15−6 ζ156 ζ153 ζ15−3 −ζ15−5 −ζ155 −ζ15−3 −ζ153 −ζ156 −ζ15−6 ζ15−2 ζ152 ζ15−4 ζ154 ζ157 ζ15−7 ζ15 ζ15−1 −ζ15−1 −ζ15 −ζ15−7 −ζ157 −ζ154 −ζ15−4 −ζ152 −ζ15−2
30.4.1h5 C 1 −1 ζ15−5 ζ155 ζ15−3 ζ153 ζ15−6 ζ156 −ζ155 −ζ15−5 −ζ156 −ζ15−6 −ζ153 −ζ15−3 ζ15−1 ζ15 ζ15−2 ζ152 ζ15−4 ζ154 ζ15−7 ζ157 −ζ157 −ζ15−7 −ζ154 −ζ15−4 −ζ152 −ζ15−2 −ζ15 −ζ15−1
30.4.1h6 C 1 −1 ζ155 ζ15−5 ζ153 ζ15−3 ζ156 ζ15−6 −ζ15−5 −ζ155 −ζ15−6 −ζ156 −ζ15−3 −ζ153 ζ15 ζ15−1 ζ152 ζ15−2 ζ154 ζ15−4 ζ157 ζ15−7 −ζ15−7 −ζ157 −ζ15−4 −ζ154 −ζ15−2 −ζ152 −ζ15−1 −ζ15
30.4.1h7 C 1 −1 ζ15−5 ζ155 ζ153 ζ15−3 ζ156 ζ15−6 −ζ155 −ζ15−5 −ζ15−6 −ζ156 −ζ15−3 −ζ153 ζ15−4 ζ154 ζ157 ζ15−7 ζ15−1 ζ15 ζ152 ζ15−2 −ζ15−2 −ζ152 −ζ15 −ζ15−1 −ζ15−7 −ζ157 −ζ154 −ζ15−4
30.4.1h8 C 1 −1 ζ155 ζ15−5 ζ15−3 ζ153 ζ15−6 ζ156 −ζ15−5 −ζ155 −ζ156 −ζ15−6 −ζ153 −ζ15−3 ζ154 ζ15−4 ζ15−7 ζ157 ζ15 ζ15−1 ζ15−2 ζ152 −ζ152 −ζ15−2 −ζ15−1 −ζ15 −ζ157 −ζ15−7 −ζ15−4 −ζ154

magma: CharacterTable(G);