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Properties

Label 30T45
Degree $30$
Order $180$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\GL(2,4)$

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Show commands: Magma

magma: G := TransitiveGroup(30, 45);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\GL(2,4)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8)(2,9)(3,7)(10,29)(11,30)(12,28)(13,25)(14,26)(15,27)(16,24)(17,22)(18,23), (1,7,15,30,18,2,8,13,28,16,3,9,14,29,17)(4,25,22,20,10,5,26,23,21,11,6,27,24,19,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$60$:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 5: None

Degree 6: None

Degree 10: $A_{5}$

Degree 15: None

Low degree siblings

15T15 x 2, 15T16, 18T90, 36T176, 45T16

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  180.19
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B 3C1 3C-1 5A1 5A2 6A1 6A-1 15A1 15A-1 15A2 15A-2
Size 1 15 1 1 20 20 20 12 12 15 15 12 12 12 12
2 P 1A 1A 3A-1 3A1 3C1 3B 3C-1 5A2 5A1 3A1 3A-1 15A-1 15A2 15A1 15A-2
3 P 1A 2A 1A 1A 1A 1A 1A 5A2 5A1 2A 2A 5A2 5A1 5A2 5A1
5 P 1A 2A 3A-1 3A1 3C1 3B 3C-1 1A 1A 6A-1 6A1 3A1 3A1 3A-1 3A-1
Type
180.19.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
180.19.1b1 C 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
180.19.1b2 C 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
180.19.3a1 R 3 −1 3 3 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −1 −1 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
180.19.3a2 R 3 −1 3 3 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −1 −1 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
180.19.3b1 C 3 −1 3ζ15−5 3ζ155 0 0 0 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ155 −ζ15−5 −1+ζ15+ζ152−ζ153+ζ154+ζ157 −ζ15−ζ154 −1+ζ15+ζ154−ζ155 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157
180.19.3b2 C 3 −1 3ζ155 3ζ15−5 0 0 0 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ15−5 −ζ155 −ζ15−ζ154 −1+ζ15+ζ152−ζ153+ζ154+ζ157 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 −1+ζ15+ζ154−ζ155
180.19.3b3 C 3 −1 3ζ15−5 3ζ155 0 0 0 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ155 −ζ15−5 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 −1+ζ15+ζ154−ζ155 −ζ15−ζ154 −1+ζ15+ζ152−ζ153+ζ154+ζ157
180.19.3b4 C 3 −1 3ζ155 3ζ15−5 0 0 0 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ15−5 −ζ155 −1+ζ15+ζ154−ζ155 1−ζ15−ζ152+ζ153−ζ154+ζ155−ζ157 −1+ζ15+ζ152−ζ153+ζ154+ζ157 −ζ15−ζ154
180.19.4a R 4 0 4 4 1 1 1 −1 −1 0 0 −1 −1 −1 −1
180.19.4b1 C 4 0 4ζ3−1 4ζ3 1 ζ3−1 ζ3 −1 −1 0 0 −ζ3 −ζ3−1 −ζ3−1 −ζ3
180.19.4b2 C 4 0 4ζ3 4ζ3−1 1 ζ3 ζ3−1 −1 −1 0 0 −ζ3−1 −ζ3 −ζ3 −ζ3−1
180.19.5a R 5 1 5 5 −1 −1 −1 0 0 1 1 0 0 0 0
180.19.5b1 C 5 1 5ζ3−1 5ζ3 −1 −ζ3−1 −ζ3 0 0 ζ3 ζ3−1 0 0 0 0
180.19.5b2 C 5 1 5ζ3 5ζ3−1 −1 −ζ3 −ζ3−1 0 0 ζ3−1 ζ3 0 0 0 0

magma: CharacterTable(G);