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Properties

Label 45T6
Degree $45$
Order $90$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3:D_{15}$

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Show commands: Magma

magma: G := TransitiveGroup(45, 6);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$ x 4
$10$:  $D_{5}$
$18$:  $C_3^2:C_2$
$30$:  $D_{15}$ x 4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 4

Degree 5: $D_{5}$

Degree 9: $C_3^2:C_2$

Degree 15: $D_{15}$ x 4

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $90=2 \cdot 3^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  90.9
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B 3C 3D 5A1 5A2 15A1 15A2 15A4 15A7 15B1 15B2 15B4 15B7 15C1 15C2 15C4 15C7 15D1 15D2 15D4 15D7
Size 1 45 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 3D 3A 3C 3B 5A2 5A1 15B2 15D1 15B1 15D7 15A4 15D4 15C7 15C2 15B7 15C1 15A1 15D2 15A2 15C4 15A7 15B4
3 P 1A 2A 1A 1A 1A 1A 5A2 5A1 5A1 5A1 5A2 5A2 5A2 5A1 5A1 5A1 5A1 5A2 5A2 5A2 5A1 5A2 5A1 5A2
5 P 1A 2A 3D 3A 3C 3B 1A 1A 3B 3D 3B 3D 3A 3D 3C 3C 3B 3C 3A 3D 3A 3C 3A 3B
Type
90.9.1a 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
90.9.1b 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
90.9.2a R 2 0 −1 −1 −1 2 2 2 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 2 2 2 2
90.9.2b R 2 0 −1 −1 2 −1 2 2 −1 −1 −1 −1 −1 −1 −1 −1 2 2 2 2 −1 −1 −1 −1
90.9.2c R 2 0 −1 2 −1 −1 2 2 −1 −1 −1 −1 2 2 2 2 −1 −1 −1 −1 −1 −1 −1 −1
90.9.2d R 2 0 2 −1 −1 −1 2 2 2 2 2 2 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
90.9.2e1 R 2 0 2 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
90.9.2e2 R 2 0 2 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
90.9.2f1 R 2 0 −1 −1 −1 2 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153
90.9.2f2 R 2 0 −1 −1 −1 2 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153
90.9.2f3 R 2 0 −1 −1 −1 2 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156
90.9.2f4 R 2 0 −1 −1 −1 2 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156
90.9.2g1 R 2 0 −1 −1 2 −1 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157
90.9.2g2 R 2 0 −1 −1 2 −1 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152
90.9.2g3 R 2 0 −1 −1 2 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154
90.9.2g4 R 2 0 −1 −1 2 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15
90.9.2h1 R 2 0 −1 2 −1 −1 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152
90.9.2h2 R 2 0 −1 2 −1 −1 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157
90.9.2h3 R 2 0 −1 2 −1 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15
90.9.2h4 R 2 0 −1 2 −1 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154
90.9.2i1 R 2 0 2 −1 −1 −1 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157
90.9.2i2 R 2 0 2 −1 −1 −1 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152
90.9.2i3 R 2 0 2 −1 −1 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154
90.9.2i4 R 2 0 2 −1 −1 −1 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15

magma: CharacterTable(G);