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Properties

Label 30T17
Degree $30$
Order $120$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{30}:C_4$

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magma: G := TransitiveGroup(30, 17);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $17$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{30}:C_4$
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:  (3,6,9,18)(4,5,10,17)(7,14,25,19)(8,13,26,20)(11,21)(12,22)(15,29,27,24)(16,30,28,23), (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)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$, $C_3 : C_4$ x 2
$20$:  $F_5$
$24$:  24T6
$40$:  $F_{5}\times C_2$
$60$:  $C_{15} : C_4$

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: $F_{5}\times C_2$

Degree 15: $C_{15} : C_4$

Low degree siblings

30T17

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  120.41
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 5A 6A 6B 6C 10A 15A1 15A-1 30A1 30A-1
Size 1 1 5 5 2 15 15 15 15 4 2 10 10 4 4 4 4 4
2 P 1A 1A 1A 1A 3A 2B 2B 2B 2B 5A 3A 3A 3A 5A 15A1 15A-1 15A1 15A-1
3 P 1A 2A 2B 2C 1A 4B1 4B-1 4A1 4A-1 5A 2A 2C 2B 10A 5A 5A 10A 10A
5 P 1A 2A 2B 2C 3A 4B-1 4B1 4A-1 4A1 1A 6A 6B 6C 2A 3A 3A 6A 6A
Type
120.41.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
120.41.1b R 1 −1 1 −1 1 −1 −1 1 1 1 −1 −1 1 −1 1 1 −1 −1
120.41.1c R 1 −1 1 −1 1 1 1 −1 −1 1 −1 −1 1 −1 1 1 −1 −1
120.41.1d R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 1
120.41.1e1 C 1 −1 −1 1 1 −i i −i i 1 −1 1 −1 −1 1 1 −1 −1
120.41.1e2 C 1 −1 −1 1 1 i −i i −i 1 −1 1 −1 −1 1 1 −1 −1
120.41.1f1 C 1 1 −1 −1 1 −i i i −i 1 1 −1 −1 1 1 1 1 1
120.41.1f2 C 1 1 −1 −1 1 i −i −i i 1 1 −1 −1 1 1 1 1 1
120.41.2a R 2 2 2 2 −1 0 0 0 0 2 −1 −1 −1 2 −1 −1 −1 −1
120.41.2b R 2 −2 2 −2 −1 0 0 0 0 2 1 1 −1 −2 −1 −1 1 1
120.41.2c S 2 −2 −2 2 −1 0 0 0 0 2 1 −1 1 −2 −1 −1 1 1
120.41.2d S 2 2 −2 −2 −1 0 0 0 0 2 −1 1 1 2 −1 −1 −1 −1
120.41.4a R 4 4 0 0 4 0 0 0 0 −1 4 0 0 −1 −1 −1 −1 −1
120.41.4b R 4 −4 0 0 4 0 0 0 0 −1 −4 0 0 1 −1 −1 1 1
120.41.4c1 C 4 4 0 0 −2 0 0 0 0 −1 −2 0 0 −1 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157
120.41.4c2 C 4 4 0 0 −2 0 0 0 0 −1 −2 0 0 −1 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157
120.41.4d1 C 4 −4 0 0 −2 0 0 0 0 −1 2 0 0 1 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 −2+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 1−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157
120.41.4d2 C 4 −4 0 0 −2 0 0 0 0 −1 2 0 0 1 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 1−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −2+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157

magma: CharacterTable(G);