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Properties

Label 33T7
Degree $33$
Order $330$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{33}:C_{10}$

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magma: G := TransitiveGroup(33, 7);
 

Group action invariants

Degree $n$:  $33$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{33}:C_{10}$
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:  (1,19,8,6,28,2,21,9,5,29)(3,20,7,4,30)(10,13,23,18,32,12,14,22,16,31)(11,15,24,17,33)(25,26), (1,11,19,28,4,14,22,33,9,18,27,2,12,20,29,5,15,23,31,7,16,25,3,10,21,30,6,13,24,32,8,17,26)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$
$30$:  $S_3 \times C_5$
$55$:  $C_{11}:C_5$
$110$:  22T5

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 11: $C_{11}:C_5$

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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ $11$ $5$ $( 4,11,30,17,15)( 5,12,28,18,13)( 6,10,29,16,14)( 7,20,24,33,27) ( 8,21,22,31,25)( 9,19,23,32,26)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ $11$ $5$ $( 4,15,17,30,11)( 5,13,18,28,12)( 6,14,16,29,10)( 7,27,33,24,20) ( 8,25,31,22,21)( 9,26,32,23,19)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ $11$ $5$ $( 4,17,11,15,30)( 5,18,12,13,28)( 6,16,10,14,29)( 7,33,20,27,24) ( 8,31,21,25,22)( 9,32,19,26,23)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ $11$ $5$ $( 4,30,15,11,17)( 5,28,13,12,18)( 6,29,14,10,16)( 7,24,27,20,33) ( 8,22,25,21,31)( 9,23,26,19,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 6)( 7, 9)(10,11)(14,15)(16,17)(19,20)(23,24)(26,27)(29,30)(32,33)$
$ 10, 10, 5, 5, 2, 1 $ $33$ $10$ $( 2, 3)( 4,10,30,16,15, 6,11,29,17,14)( 5,12,28,18,13)( 7,19,24,32,27, 9,20, 23,33,26)( 8,21,22,31,25)$
$ 10, 10, 5, 5, 2, 1 $ $33$ $10$ $( 2, 3)( 4,14,17,29,11, 6,15,16,30,10)( 5,13,18,28,12)( 7,26,33,23,20, 9,27, 32,24,19)( 8,25,31,22,21)$
$ 10, 10, 5, 5, 2, 1 $ $33$ $10$ $( 2, 3)( 4,16,11,14,30, 6,17,10,15,29)( 5,18,12,13,28)( 7,32,20,26,24, 9,33, 19,27,23)( 8,31,21,25,22)$
$ 10, 10, 5, 5, 2, 1 $ $33$ $10$ $( 2, 3)( 4,29,15,10,17, 6,30,14,11,16)( 5,28,13,12,18)( 7,23,27,19,33, 9,24, 26,20,32)( 8,22,25,21,31)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)$
$ 15, 15, 3 $ $22$ $15$ $( 1, 2, 3)( 4,12,29,17,13, 6,11,28,16,15, 5,10,30,18,14)( 7,21,23,33,25, 9,20, 22,32,27, 8,19,24,31,26)$
$ 15, 15, 3 $ $22$ $15$ $( 1, 2, 3)( 4,13,16,30,12, 6,15,18,29,11, 5,14,17,28,10)( 7,25,32,24,21, 9,27, 31,23,20, 8,26,33,22,19)$
$ 15, 15, 3 $ $22$ $15$ $( 1, 2, 3)( 4,18,10,15,28, 6,17,12,14,30, 5,16,11,13,29)( 7,31,19,27,22, 9,33, 21,26,24, 8,32,20,25,23)$
$ 15, 15, 3 $ $22$ $15$ $( 1, 2, 3)( 4,28,14,11,18, 6,30,13,10,17, 5,29,15,12,16)( 7,22,26,20,31, 9,24, 25,19,33, 8,23,27,21,32)$
$ 33 $ $10$ $33$ $( 1, 4, 9,12,15,16,21,24,26,28,33, 2, 5, 7,10,13,17,19,22,27,29,31, 3, 6, 8, 11,14,18,20,23,25,30,32)$
$ 22, 11 $ $15$ $22$ $( 1, 4, 8,11,13,17,21,24,25,30,31, 3, 5, 7,12,15,18,20,22,27,28,33) ( 2, 6, 9,10,14,16,19,23,26,29,32)$
$ 11, 11, 11 $ $5$ $11$ $( 1, 5, 8,12,13,18,21,22,25,28,31)( 2, 6, 9,10,14,16,19,23,26,29,32) ( 3, 4, 7,11,15,17,20,24,27,30,33)$
$ 33 $ $10$ $33$ $( 1, 7,14,21,27,32, 5,11,16,22,30, 2, 8,15,19,25,33, 6,12,17,23,28, 3, 9,13, 20,26,31, 4,10,18,24,29)$
$ 22, 11 $ $15$ $22$ $( 1, 7,13,20,25,33, 5,11,18,24,28, 3, 8,15,21,27,31, 4,12,17,22,30) ( 2, 9,14,19,26,32, 6,10,16,23,29)$
$ 11, 11, 11 $ $5$ $11$ $( 1, 8,13,21,25,31, 5,12,18,22,28)( 2, 9,14,19,26,32, 6,10,16,23,29) ( 3, 7,15,20,27,33, 4,11,17,24,30)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $330=2 \cdot 3 \cdot 5 \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  330.2
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 11A1 11A-1 15A1 15A-1 15A2 15A-2 22A1 22A-1 33A1 33A-1
Size 1 3 2 11 11 11 11 33 33 33 33 5 5 22 22 22 22 15 15 10 10
2 P 1A 1A 3A 5A1 5A2 5A-1 5A-2 5A-1 5A1 5A2 5A-2 11A-1 11A1 15A2 15A-2 15A-1 15A1 11A1 11A-1 33A-1 33A1
3 P 1A 2A 1A 5A-1 5A-2 5A1 5A2 10A3 10A-3 10A-1 10A1 11A1 11A-1 5A1 5A-1 5A2 5A-2 22A1 22A-1 11A1 11A-1
5 P 1A 2A 3A 1A 1A 1A 1A 2A 2A 2A 2A 11A1 11A-1 3A 3A 3A 3A 22A1 22A-1 33A1 33A-1
11 P 1A 2A 3A 5A-2 5A1 5A2 5A-1 10A1 10A-1 10A3 10A-3 1A 1A 15A1 15A-1 15A2 15A-2 2A 2A 3A 3A
Type
330.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
330.2.1b R 1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 1 1
330.2.1c1 C 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52 1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1
330.2.1c2 C 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ52 ζ5−2 1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1
330.2.1c3 C 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5−1 ζ5 1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1
330.2.1c4 C 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1
330.2.1d1 C 1 −1 1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52 1 1 ζ5 ζ5−1 ζ52 ζ5−2 −1 −1 1 1
330.2.1d2 C 1 −1 1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2 1 1 ζ5−1 ζ5 ζ5−2 ζ52 −1 −1 1 1
330.2.1d3 C 1 −1 1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5 1 1 ζ5−2 ζ52 ζ5 ζ5−1 −1 −1 1 1
330.2.1d4 C 1 −1 1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 −1 −1 1 1
330.2.2a R 2 0 −1 2 2 2 2 0 0 0 0 2 2 −1 −1 −1 −1 0 0 −1 −1
330.2.2b1 C 2 0 −1 2ζ5−2 2ζ52 2ζ5 2ζ5−1 0 0 0 0 2 2 −ζ5 −ζ5−1 −ζ52 −ζ5−2 0 0 −1 −1
330.2.2b2 C 2 0 −1 2ζ52 2ζ5−2 2ζ5−1 2ζ5 0 0 0 0 2 2 −ζ5−1 −ζ5 −ζ5−2 −ζ52 0 0 −1 −1
330.2.2b3 C 2 0 −1 2ζ5−1 2ζ5 2ζ5−2 2ζ52 0 0 0 0 2 2 −ζ5−2 −ζ52 −ζ5 −ζ5−1 0 0 −1 −1
330.2.2b4 C 2 0 −1 2ζ5 2ζ5−1 2ζ52 2ζ5−2 0 0 0 0 2 2 −ζ52 −ζ5−2 −ζ5−1 −ζ5 0 0 −1 −1
330.2.5a1 C 5 5 5 0 0 0 0 0 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 0 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
330.2.5a2 C 5 5 5 0 0 0 0 0 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 0 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
330.2.5b1 C 5 −5 5 0 0 0 0 0 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 0 0 0 0 −ζ11−2−ζ11−ζ113−ζ114−ζ115 ζ11−2+1+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
330.2.5b2 C 5 −5 5 0 0 0 0 0 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 0 0 0 0 ζ11−2+1+ζ11+ζ113+ζ114+ζ115 −ζ11−2−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
330.2.10a1 C 10 0 −5 0 0 0 0 0 0 0 0 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 0 0 0 0 0 0 ζ11−2+1+ζ11+ζ113+ζ114+ζ115 −ζ11−2−ζ11−ζ113−ζ114−ζ115
330.2.10a2 C 10 0 −5 0 0 0 0 0 0 0 0 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 0 0 0 0 0 0 −ζ11−2−ζ11−ζ113−ζ114−ζ115 ζ11−2+1+ζ11+ζ113+ζ114+ζ115

magma: CharacterTable(G);