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Properties

Label 36T550
36T550 1 12 1->12 34 1->34 2 11 2->11 33 2->33 3 7 3->7 35 3->35 4 8 4->8 36 4->36 5 9 5->9 31 5->31 6 10 6->10 32 6->32 17 7->17 25 7->25 18 8->18 26 8->26 14 9->14 28 9->28 13 10->13 27 10->27 15 11->15 30 11->30 16 12->16 29 12->29 19 13->19 20 14->20 21 15->21 22 16->22 24 17->24 23 18->23 19->5 19->27 20->6 20->28 21->1 21->30 22->2 22->29 23->3 23->26 24->4 24->25 25->15 26->16 27->17 28->18 29->13 30->14 31->34 32->33 33->36 34->35 35->31 36->32
Degree $36$
Order $360$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_3\times S_5$

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Copy content magma:G := TransitiveGroup(36, 550);
 

Group invariants

Abstract group:  $C_3\times S_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $36$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $550$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $6$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,12,29,13,19,5,9,28,18,23,3,7,25,15,21)(2,11,30,14,20,6,10,27,17,24,4,8,26,16,22)(31,34,35)(32,33,36)$, $(1,34)(2,33)(3,35)(4,36)(5,31)(6,32)(7,17)(8,18)(9,14)(10,13)(11,15)(12,16)(19,27)(20,28)(21,30)(22,29)(23,26)(24,25)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$120$:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: None

Degree 6: $C_6$, $\PGL(2,5)$

Degree 9: None

Degree 12: $S_5$

Degree 18: 18T144

Low degree siblings

15T24, 18T144, 30T90, 30T98, 30T103, 45T44

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{36}$ $1$ $1$ $0$ $()$
2A $2^{18}$ $10$ $2$ $18$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5, 8)( 6, 7)(13,26)(14,25)(15,27)(16,28)(17,29)(18,30)(19,35)(20,36)(21,31)(22,32)(23,34)(24,33)$
2B $2^{12},1^{12}$ $15$ $2$ $12$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,18)( 6,17)( 7,29)( 8,30)( 9,25)(10,26)(11,27)(12,28)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1, 5, 3)( 2, 6, 4)( 7,12, 9)( 8,11,10)(13,18,15)(14,17,16)(19,23,21)(20,24,22)(25,29,28)(26,30,27)(31,35,34)(32,36,33)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,15,18)(14,16,17)(19,21,23)(20,22,24)(25,28,29)(26,27,30)(31,34,35)(32,33,36)$
3B $3^{12}$ $20$ $3$ $24$ $( 1,13, 9)( 2,14,10)( 3,15,12)( 4,16,11)( 5,18, 7)( 6,17, 8)(19,36,28)(20,35,27)(21,32,29)(22,31,30)(23,33,25)(24,34,26)$
3C1 $3^{12}$ $20$ $3$ $24$ $( 1,28,32)( 2,27,31)( 3,29,33)( 4,30,34)( 5,25,36)( 6,26,35)( 7,23,15)( 8,24,16)( 9,19,18)(10,20,17)(11,22,14)(12,21,13)$
3C-1 $3^{12}$ $20$ $3$ $24$ $( 1,32,28)( 2,31,27)( 3,33,29)( 4,34,30)( 5,36,25)( 6,35,26)( 7,15,23)( 8,16,24)( 9,18,19)(10,17,20)(11,14,22)(12,13,21)$
4A $4^{6},2^{6}$ $30$ $4$ $24$ $( 1,26,13,10)( 2,25,14, 9)( 3,27,15,11)( 4,28,16,12)( 5,30,18, 8)( 6,29,17, 7)(19,20)(21,22)(23,24)(31,32)(33,34)(35,36)$
5A $5^{6},1^{6}$ $24$ $5$ $24$ $( 7,18,29,21,32)( 8,17,30,22,31)( 9,13,25,23,33)(10,14,26,24,34)(11,16,27,20,35)(12,15,28,19,36)$
6A1 $6^{6}$ $10$ $6$ $30$ $( 1,11, 5,10, 3, 8)( 2,12, 6, 9, 4, 7)(13,27,18,26,15,30)(14,28,17,25,16,29)(19,31,23,35,21,34)(20,32,24,36,22,33)$
6A-1 $6^{6}$ $10$ $6$ $30$ $( 1, 8, 3,10, 5,11)( 2, 7, 4, 9, 6,12)(13,30,15,26,18,27)(14,29,16,25,17,28)(19,34,21,35,23,31)(20,33,22,36,24,32)$
6B1 $6^{4},3^{4}$ $15$ $6$ $28$ $( 1,15, 5,13, 3,18)( 2,16, 6,14, 4,17)( 7,25,12,29, 9,28)( 8,26,11,30,10,27)(19,21,23)(20,22,24)(31,34,35)(32,33,36)$
6B-1 $6^{4},3^{4}$ $15$ $6$ $28$ $( 1,18, 3,13, 5,15)( 2,17, 4,14, 6,16)( 7,28, 9,29,12,25)( 8,27,10,30,11,26)(19,23,21)(20,24,22)(31,35,34)(32,36,33)$
6C $6^{6}$ $20$ $6$ $30$ $( 1,26,13,24, 9,34)( 2,25,14,23,10,33)( 3,27,15,20,12,35)( 4,28,16,19,11,36)( 5,30,18,22, 7,31)( 6,29,17,21, 8,32)$
6D1 $6^{6}$ $20$ $6$ $30$ $( 1, 8,28,24,32,16)( 2, 7,27,23,31,15)( 3,10,29,20,33,17)( 4, 9,30,19,34,18)( 5,11,25,22,36,14)( 6,12,26,21,35,13)$
6D-1 $6^{6}$ $20$ $6$ $30$ $( 1,16,32,24,28, 8)( 2,15,31,23,27, 7)( 3,17,33,20,29,10)( 4,18,34,19,30, 9)( 5,14,36,22,25,11)( 6,13,35,21,26,12)$
12A1 $12^{2},6^{2}$ $30$ $12$ $32$ $( 1, 8,15,26, 5,11,13,30, 3,10,18,27)( 2, 7,16,25, 6,12,14,29, 4, 9,17,28)(19,24,21,20,23,22)(31,36,34,32,35,33)$
12A-1 $12^{2},6^{2}$ $30$ $12$ $32$ $( 1,27,18,10, 3,30,13,11, 5,26,15, 8)( 2,28,17, 9, 4,29,14,12, 6,25,16, 7)(19,22,23,20,21,24)(31,33,35,32,34,36)$
15A1 $15^{2},3^{2}$ $24$ $15$ $32$ $( 1, 3, 5)( 2, 4, 6)( 7,25,36,18,23,12,29,33,15,21, 9,28,32,13,19)( 8,26,35,17,24,11,30,34,16,22,10,27,31,14,20)$
15A-1 $15^{2},3^{2}$ $24$ $15$ $32$ $( 1, 5, 3)( 2, 6, 4)( 7,19,13,32,28, 9,21,15,33,29,12,23,18,36,25)( 8,20,14,31,27,10,22,16,34,30,11,24,17,35,26)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A1 3A-1 3B 3C1 3C-1 4A 5A 6A1 6A-1 6B1 6B-1 6C 6D1 6D-1 12A1 12A-1 15A1 15A-1
Size 1 10 15 1 1 20 20 20 30 24 10 10 15 15 20 20 20 30 30 24 24
2 P 1A 1A 1A 3A-1 3A1 3B 3C-1 3C1 2B 5A 3A1 3A-1 3A1 3A-1 3B 3C1 3C-1 6B1 6B-1 15A-1 15A1
3 P 1A 2A 2B 1A 1A 1A 1A 1A 4A 5A 2A 2A 2B 2B 2A 2A 2A 4A 4A 5A 5A
5 P 1A 2A 2B 3A-1 3A1 3B 3C-1 3C1 4A 1A 6A-1 6A1 6B-1 6B1 6C 6D-1 6D1 12A-1 12A1 3A1 3A-1
Type
360.119.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
360.119.1b R 1 −1 1 1 1 1 1 1 −1 1 −1 −1 1 1 −1 −1 −1 −1 −1 1 1
360.119.1c1 C 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
360.119.1c2 C 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
360.119.1d1 C 1 −1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 −1 1 −ζ3 −ζ3−1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1
360.119.1d2 C 1 −1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 −1 1 −ζ3−1 −ζ3 ζ3−1 ζ3 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3
360.119.4a R 4 2 0 4 4 1 1 1 0 −1 2 2 0 0 −1 −1 −1 0 0 −1 −1
360.119.4b R 4 −2 0 4 4 1 1 1 0 −1 −2 −2 0 0 1 1 1 0 0 −1 −1
360.119.4c1 C 4 2 0 4ζ3−1 4ζ3 1 ζ3 ζ3−1 0 −1 2ζ3 2ζ3−1 0 0 −1 −ζ3−1 −ζ3 0 0 −ζ3 −ζ3−1
360.119.4c2 C 4 2 0 4ζ3 4ζ3−1 1 ζ3−1 ζ3 0 −1 2ζ3−1 2ζ3 0 0 −1 −ζ3 −ζ3−1 0 0 −ζ3−1 −ζ3
360.119.4d1 C 4 −2 0 4ζ3−1 4ζ3 1 ζ3 ζ3−1 0 −1 −2ζ3 −2ζ3−1 0 0 1 ζ3−1 ζ3 0 0 −ζ3 −ζ3−1
360.119.4d2 C 4 −2 0 4ζ3 4ζ3−1 1 ζ3−1 ζ3 0 −1 −2ζ3−1 −2ζ3 0 0 1 ζ3 ζ3−1 0 0 −ζ3−1 −ζ3
360.119.5a R 5 −1 1 5 5 −1 −1 −1 1 0 −1 −1 1 1 −1 −1 −1 1 1 0 0
360.119.5b R 5 1 1 5 5 −1 −1 −1 −1 0 1 1 1 1 1 1 1 −1 −1 0 0
360.119.5c1 C 5 −1 1 5ζ3−1 5ζ3 −1 −ζ3 −ζ3−1 1 0 −ζ3 −ζ3−1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3 ζ3−1 ζ3 0 0
360.119.5c2 C 5 −1 1 5ζ3 5ζ3−1 −1 −ζ3−1 −ζ3 1 0 −ζ3−1 −ζ3 ζ3−1 ζ3 −1 −ζ3 −ζ3−1 ζ3 ζ3−1 0 0
360.119.5d1 C 5 1 1 5ζ3−1 5ζ3 −1 −ζ3 −ζ3−1 −1 0 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3−1 −ζ3 0 0
360.119.5d2 C 5 1 1 5ζ3 5ζ3−1 −1 −ζ3−1 −ζ3 −1 0 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3 −ζ3−1 0 0
360.119.6a R 6 0 −2 6 6 0 0 0 0 1 0 0 −2 −2 0 0 0 0 0 1 1
360.119.6b1 C 6 0 −2 6ζ3−1 6ζ3 0 0 0 0 1 0 0 −2ζ3 −2ζ3−1 0 0 0 0 0 ζ3 ζ3−1
360.119.6b2 C 6 0 −2 6ζ3 6ζ3−1 0 0 0 0 1 0 0 −2ζ3−1 −2ζ3 0 0 0 0 0 ζ3−1 ζ3

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed