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Properties

Label 30T48
Degree $30$
Order $180$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_3\times C_{15}):C_4$

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Show commands: Magma

magma: G := TransitiveGroup(30, 48);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $48$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(C_3\times C_{15}):C_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8,15,4,11,3,7,14,6,10,2,9,13,5,12)(16,24,29,19,27,17,22,30,20,25,18,23,28,21,26), (1,26,3,25)(2,27)(4,23,6,22)(5,24)(7,20,9,19)(8,21)(10,17,12,16)(11,18)(13,29,15,28)(14,30)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$10$:  $D_{5}$
$20$:  20T2
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $D_{5}$

Degree 6: $C_3^2:C_4$

Degree 10: $D_5$

Degree 15: None

Low degree siblings

30T48, 45T26

Siblings are shown 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)$
$ 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)(16,19,22,25,28) (17,20,23,26,29)(18,21,24,27,30)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)(16,20,24,25,29,18,19,23,27, 28,17,21,22,26,30)$
$ 10, 10, 5, 5 $ $18$ $10$ $( 1, 4, 7,10,13)( 2, 6, 8,12,14, 3, 5, 9,11,15)(16,19,22,25,28) (17,21,23,27,29,18,20,24,26,30)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 5, 9,10,14, 3, 4, 8,12,13, 2, 6, 7,11,15)(16,19,22,25,28)(17,20,23,26,29) (18,21,24,27,30)$
$ 15, 15 $ $4$ $15$ $( 1, 5, 9,10,14, 3, 4, 8,12,13, 2, 6, 7,11,15)(16,20,24,25,29,18,19,23,27,28, 17,21,22,26,30)$
$ 15, 15 $ $4$ $15$ $( 1, 5, 9,10,14, 3, 4, 8,12,13, 2, 6, 7,11,15)(16,21,23,25,30,17,19,24,26,28, 18,20,22,27,29)$
$ 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)(16,22,28,19,25) (17,23,29,20,26)(18,24,30,21,27)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)(16,23,30,19,26,18,22,29,21, 25,17,24,28,20,27)$
$ 10, 10, 5, 5 $ $18$ $10$ $( 1, 7,13, 4,10)( 2, 9,14, 6,11, 3, 8,15, 5,12)(16,22,28,19,25) (17,24,29,21,26,18,23,30,20,27)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 8,15, 4,11, 3, 7,14, 6,10, 2, 9,13, 5,12)(16,22,28,19,25)(17,23,29,20,26) (18,24,30,21,27)$
$ 15, 15 $ $4$ $15$ $( 1, 8,15, 4,11, 3, 7,14, 6,10, 2, 9,13, 5,12)(16,23,30,19,26,18,22,29,21,25, 17,24,28,20,27)$
$ 15, 15 $ $4$ $15$ $( 1, 8,15, 4,11, 3, 7,14, 6,10, 2, 9,13, 5,12)(16,24,29,19,27,17,22,30,20,25, 18,23,28,21,26)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2 $ $45$ $4$ $( 1,16)( 2,17, 3,18)( 4,28)( 5,29, 6,30)( 7,25)( 8,26, 9,27)(10,22) (11,23,12,24)(13,19)(14,20,15,21)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2 $ $45$ $4$ $( 1,16)( 2,18, 3,17)( 4,28)( 5,30, 6,29)( 7,25)( 8,27, 9,26)(10,22) (11,24,12,23)(13,19)(14,21,15,20)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  180.24
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B 4A1 4A-1 5A1 5A2 10A1 10A3 15A1 15A-1 15A2 15A-2 15B1 15B-1 15B2 15B-2
Size 1 9 4 4 45 45 2 2 18 18 4 4 4 4 4 4 4 4
2 P 1A 1A 3A 3B 2A 2A 5A2 5A1 5A1 5A2 15A-1 15B1 15A2 15B-2 15B-1 15A1 15B2 15A-2
3 P 1A 2A 1A 1A 4A-1 4A1 5A2 5A1 10A3 10A1 5A2 5A2 5A1 5A1 5A2 5A2 5A1 5A1
5 P 1A 2A 3A 3B 4A1 4A-1 1A 1A 2A 2A 3A 3B 3A 3B 3B 3A 3B 3A
Type
180.24.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
180.24.1b R 1 1 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1
180.24.1c1 C 1 −1 1 1 −i i 1 1 −1 −1 1 1 1 1 1 1 1 1
180.24.1c2 C 1 −1 1 1 i −i 1 1 −1 −1 1 1 1 1 1 1 1 1
180.24.2a1 R 2 2 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
180.24.2a2 R 2 2 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
180.24.2b1 S 2 −2 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
180.24.2b2 S 2 −2 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
180.24.4a R 4 0 −2 1 0 0 4 4 0 0 −2 −2 −2 −2 1 1 1 1
180.24.4b R 4 0 1 −2 0 0 4 4 0 0 1 1 1 1 −2 −2 −2 −2
180.24.4c1 C 4 0 −2 1 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −2ζ5−2−2−3ζ5−2ζ52 ζ5−2+1+3ζ5+ζ52 2ζ5−2−ζ52 −ζ5−2+2ζ52
180.24.4c2 C 4 0 −2 1 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 ζ5−2+1+3ζ5+ζ52 −2ζ5−2−2−3ζ5−2ζ52 −ζ5−2+2ζ52 2ζ5−2−ζ52
180.24.4c3 C 4 0 −2 1 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 2ζ5−2−ζ52 −ζ5−2+2ζ52 ζ5−2+1+3ζ5+ζ52 −2ζ5−2−2−3ζ5−2ζ52
180.24.4c4 C 4 0 −2 1 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2+2ζ52 2ζ5−2−ζ52 −2ζ5−2−2−3ζ5−2ζ52 ζ5−2+1+3ζ5+ζ52
180.24.4d1 C 4 0 1 −2 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 −2ζ5−2−2−3ζ5−2ζ52 ζ5−2+1+3ζ5+ζ52 2ζ5−2−ζ52 −ζ5−2+2ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
180.24.4d2 C 4 0 1 −2 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 ζ5−2+1+3ζ5+ζ52 −2ζ5−2−2−3ζ5−2ζ52 −ζ5−2+2ζ52 2ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
180.24.4d3 C 4 0 1 −2 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 2ζ5−2−ζ52 −ζ5−2+2ζ52 ζ5−2+1+3ζ5+ζ52 −2ζ5−2−2−3ζ5−2ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
180.24.4d4 C 4 0 1 −2 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 −ζ5−2+2ζ52 2ζ5−2−ζ52 −2ζ5−2−2−3ζ5−2ζ52 ζ5−2+1+3ζ5+ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5

magma: CharacterTable(G);