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Properties

Label 45T16
Degree $45$
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(45, 16);
 

Group action invariants

Degree $n$:  $45$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $16$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\GL(2,4)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $9$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,22,18)(2,23,17)(3,24,16)(4,38,45)(5,39,43)(6,37,44)(7,20,33)(8,19,32)(9,21,31)(10,42,15)(11,40,13)(12,41,14)(25,35,30)(26,36,29)(27,34,28), (1,44,32,34,25,2,45,31,36,27,3,43,33,35,26)(4,16,13,7,37,5,18,14,8,38,6,17,15,9,39)(10,24,29,20,41,11,22,30,19,42,12,23,28,21,40)
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 3: $C_3$

Degree 5: $A_5$

Degree 9: None

Degree 15: $A_5$, $\GL(2,4)$ x 2, $\GL(2,4)$

Low degree siblings

15T15 x 2, 15T16, 18T90, 30T45, 36T176

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{45}$ $1$ $1$ $()$
$2^{18},1^{9}$ $15$ $2$ $( 4,15)( 5,13)( 6,14)( 7,38)( 8,39)( 9,37)(10,28)(11,29)(12,30)(19,40)(20,42) (21,41)(25,44)(26,43)(27,45)(31,35)(32,36)(33,34)$
$3^{15}$ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,21,20) (22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,38,39)(40,41,42) (43,44,45)$
$6^{6},3^{3}$ $15$ $6$ $( 1, 2, 3)( 4,13, 6,15, 5,14)( 7,39, 9,38, 8,37)(10,29,12,28,11,30)(16,18,17) (19,41,20,40,21,42)(22,23,24)(25,45,26,44,27,43)(31,34,32,35,33,36)$
$3^{15}$ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,17,18)(19,20,21) (22,24,23)(25,26,27)(28,30,29)(31,32,33)(34,35,36)(37,39,38)(40,42,41) (43,45,44)$
$6^{6},3^{3}$ $15$ $6$ $( 1, 3, 2)( 4,14, 5,15, 6,13)( 7,37, 8,38, 9,39)(10,30,11,28,12,29)(16,17,18) (19,42,21,40,20,41)(22,24,23)(25,43,27,44,26,45)(31,36,33,35,32,34)$
$5^{9}$ $12$ $5$ $( 1, 4,20,42,15)( 2, 5,19,40,13)( 3, 6,21,41,14)( 7,38,45,22,27) ( 8,39,43,23,26)( 9,37,44,24,25)(10,33,34,28,18)(11,32,36,29,17) (12,31,35,30,16)$
$5^{9}$ $12$ $5$ $( 1, 4,28,38,33)( 2, 5,29,39,32)( 3, 6,30,37,31)( 7,18,10,45,42) ( 8,17,11,43,40)( 9,16,12,44,41)(13,36,19,23,26)(14,35,21,24,25) (15,34,20,22,27)$
$3^{15}$ $20$ $3$ $( 1, 4,34)( 2, 5,36)( 3, 6,35)( 7,33,20)( 8,32,19)( 9,31,21)(10,38,18) (11,39,17)(12,37,16)(13,29,43)(14,30,44)(15,28,45)(22,27,42)(23,26,40) (24,25,41)$
$15^{3}$ $12$ $15$ $( 1, 5,21,42,13, 3, 4,19,41,15, 2, 6,20,40,14)( 7,39,44,22,26, 9,38,43,24,27, 8,37,45,23,25)(10,32,35,28,17,12,33,36,30,18,11,31,34,29,16)$
$15^{3}$ $12$ $15$ $( 1, 5,30,38,32, 3, 4,29,37,33, 2, 6,28,39,31)( 7,17,12,45,40, 9,18,11,44,42, 8,16,10,43,41)(13,35,20,23,25,15,36,21,22,26,14,34,19,24,27)$
$3^{15}$ $20$ $3$ $( 1, 5,35)( 2, 6,34)( 3, 4,36)( 7,32,21)( 8,31,20)( 9,33,19)(10,39,16) (11,37,18)(12,38,17)(13,30,45)(14,28,43)(15,29,44)(22,26,41)(23,25,42) (24,27,40)$
$15^{3}$ $12$ $15$ $( 1, 6,19,42,14, 2, 4,21,40,15, 3, 5,20,41,13)( 7,37,43,22,25, 8,38,44,23,27, 9,39,45,24,26)(10,31,36,28,16,11,33,35,29,18,12,32,34,30,17)$
$15^{3}$ $12$ $15$ $( 1, 6,29,38,31, 2, 4,30,39,33, 3, 5,28,37,32)( 7,16,11,45,41, 8,18,12,43,42, 9,17,10,44,40)(13,34,21,23,27,14,36,20,24,26,15,35,19,22,25)$
$3^{15}$ $20$ $3$ $( 1, 6,36)( 2, 4,35)( 3, 5,34)( 7,31,19)( 8,33,21)( 9,32,20)(10,37,17) (11,38,16)(12,39,18)(13,28,44)(14,29,45)(15,30,43)(22,25,40)(23,27,41) (24,26,42)$

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);