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Properties

Label 33T6
Degree $33$
Order $165$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{11}:C_{15}$

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Show commands: Magma

magma: G := TransitiveGroup(33, 6);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$5$:  $C_5$
$15$:  $C_{15}$
$55$:  $C_{11}:C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $165=3 \cdot 5 \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  165.1
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 5A1 5A-1 5A2 5A-2 11A1 11A-1 15A1 15A-1 15A2 15A-2 15A4 15A-4 15A7 15A-7 33A1 33A-1 33A5 33A-5
Size 1 1 1 11 11 11 11 5 5 11 11 11 11 11 11 11 11 5 5 5 5
3 P 1A 3A-1 3A1 5A2 5A-2 5A-1 5A1 11A-1 11A1 15A1 15A-4 15A2 15A-2 15A7 15A-7 15A-1 15A4 33A-1 33A1 33A-5 33A5
5 P 1A 1A 1A 5A-2 5A2 5A1 5A-1 11A1 11A-1 5A-2 5A-2 5A1 5A-1 5A1 5A-1 5A2 5A2 11A1 11A-1 11A1 11A-1
11 P 1A 3A-1 3A1 1A 1A 1A 1A 11A1 11A-1 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A1 3A-1 33A5 33A-5 33A1 33A-1
Type
165.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
165.1.1b1 C 1 ζ3−1 ζ3 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
165.1.1b2 C 1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
165.1.1c1 C 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 1 1 1 1
165.1.1c2 C 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 1 1 1 1
165.1.1c3 C 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 1 1 1 1
165.1.1c4 C 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 1 1 1 1
165.1.1d1 C 1 ζ15−5 ζ155 ζ15−6 ζ156 ζ153 ζ15−3 1 1 ζ15−7 ζ157 ζ15 ζ15−1 ζ152 ζ15−2 ζ15−4 ζ154 ζ15−5 ζ155 ζ155 ζ15−5
165.1.1d2 C 1 ζ155 ζ15−5 ζ156 ζ15−6 ζ15−3 ζ153 1 1 ζ157 ζ15−7 ζ15−1 ζ15 ζ15−2 ζ152 ζ154 ζ15−4 ζ155 ζ15−5 ζ15−5 ζ155
165.1.1d3 C 1 ζ15−5 ζ155 ζ156 ζ15−6 ζ15−3 ζ153 1 1 ζ152 ζ15−2 ζ154 ζ15−4 ζ15−7 ζ157 ζ15−1 ζ15 ζ15−5 ζ155 ζ155 ζ15−5
165.1.1d4 C 1 ζ155 ζ15−5 ζ15−6 ζ156 ζ153 ζ15−3 1 1 ζ15−2 ζ152 ζ15−4 ζ154 ζ157 ζ15−7 ζ15 ζ15−1 ζ155 ζ15−5 ζ15−5 ζ155
165.1.1d5 C 1 ζ15−5 ζ155 ζ15−3 ζ153 ζ15−6 ζ156 1 1 ζ15−1 ζ15 ζ15−2 ζ152 ζ15−4 ζ154 ζ15−7 ζ157 ζ15−5 ζ155 ζ155 ζ15−5
165.1.1d6 C 1 ζ155 ζ15−5 ζ153 ζ15−3 ζ156 ζ15−6 1 1 ζ15 ζ15−1 ζ152 ζ15−2 ζ154 ζ15−4 ζ157 ζ15−7 ζ155 ζ15−5 ζ15−5 ζ155
165.1.1d7 C 1 ζ15−5 ζ155 ζ153 ζ15−3 ζ156 ζ15−6 1 1 ζ15−4 ζ154 ζ157 ζ15−7 ζ15−1 ζ15 ζ152 ζ15−2 ζ15−5 ζ155 ζ155 ζ15−5
165.1.1d8 C 1 ζ155 ζ15−5 ζ15−3 ζ153 ζ15−6 ζ156 1 1 ζ154 ζ15−4 ζ15−7 ζ157 ζ15 ζ15−1 ζ15−2 ζ152 ζ155 ζ15−5 ζ15−5 ζ155
165.1.5a1 C 5 5 5 0 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 0 0 0 0 0 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
165.1.5a2 C 5 5 5 0 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 0 0 0 0 0 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
165.1.5b1 C 5 5ζ33−11 5ζ3311 0 0 0 0 −1−ζ333+ζ335−ζ339−ζ3312−ζ3315+ζ3316 ζ333−ζ335+ζ339+ζ3312+ζ3315−ζ3316 0 0 0 0 0 0 0 0 −ζ33−16+ζ33−14−ζ336+ζ337+ζ3310+ζ3313 −ζ33−16+ζ33−14−1−ζ333+ζ335−ζ336+ζ337−ζ339+ζ3310−ζ3311−ζ3312+ζ3313−ζ3315+ζ3316 ζ33−16−ζ33−14+1+ζ333−ζ335+ζ336−ζ337+ζ339−ζ3310+ζ3312−ζ3313+ζ3315−ζ3316 ζ33−16−ζ33−14+1+ζ336−ζ337−ζ3310+ζ3311−ζ3313
165.1.5b2 C 5 5ζ3311 5ζ33−11 0 0 0 0 ζ333−ζ335+ζ339+ζ3312+ζ3315−ζ3316 −1−ζ333+ζ335−ζ339−ζ3312−ζ3315+ζ3316 0 0 0 0 0 0 0 0 −ζ33−16+ζ33−14−1−ζ333+ζ335−ζ336+ζ337−ζ339+ζ3310−ζ3311−ζ3312+ζ3313−ζ3315+ζ3316 −ζ33−16+ζ33−14−ζ336+ζ337+ζ3310+ζ3313 ζ33−16−ζ33−14+1+ζ336−ζ337−ζ3310+ζ3311−ζ3313 ζ33−16−ζ33−14+1+ζ333−ζ335+ζ336−ζ337+ζ339−ζ3310+ζ3312−ζ3313+ζ3315−ζ3316
165.1.5b3 C 5 5ζ33−11 5ζ3311 0 0 0 0 ζ333−ζ335+ζ339+ζ3312+ζ3315−ζ3316 −1−ζ333+ζ335−ζ339−ζ3312−ζ3315+ζ3316 0 0 0 0 0 0 0 0 ζ33−16−ζ33−14+1+ζ336−ζ337−ζ3310+ζ3311−ζ3313 ζ33−16−ζ33−14+1+ζ333−ζ335+ζ336−ζ337+ζ339−ζ3310+ζ3312−ζ3313+ζ3315−ζ3316 −ζ33−16+ζ33−14−1−ζ333+ζ335−ζ336+ζ337−ζ339+ζ3310−ζ3311−ζ3312+ζ3313−ζ3315+ζ3316 −ζ33−16+ζ33−14−ζ336+ζ337+ζ3310+ζ3313
165.1.5b4 C 5 5ζ3311 5ζ33−11 0 0 0 0 −1−ζ333+ζ335−ζ339−ζ3312−ζ3315+ζ3316 ζ333−ζ335+ζ339+ζ3312+ζ3315−ζ3316 0 0 0 0 0 0 0 0 ζ33−16−ζ33−14+1+ζ333−ζ335+ζ336−ζ337+ζ339−ζ3310+ζ3312−ζ3313+ζ3315−ζ3316 ζ33−16−ζ33−14+1+ζ336−ζ337−ζ3310+ζ3311−ζ3313 −ζ33−16+ζ33−14−ζ336+ζ337+ζ3310+ζ3313 −ζ33−16+ζ33−14−1−ζ333+ζ335−ζ336+ζ337−ζ339+ζ3310−ζ3311−ζ3312+ζ3313−ζ3315+ζ3316

magma: CharacterTable(G);