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Properties

Label 36T1041
Degree $36$
Order $648$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_3\times C_6).D_{18}$

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Copy content magma:G := TransitiveGroup(36, 1041);
 

Group invariants

Abstract group:  $(C_3\times C_6).D_{18}$
Copy content magma:IdentifyGroup(G);
 
Order:  $648=2^{3} \cdot 3^{4}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
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 $t$:  $1041$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,23,2,24)(3,20,4,19)(5,21,6,22)(7,31,8,32)(9,36,10,35)(11,34,12,33)(13,29,14,30)(15,25,16,26)(17,27,18,28)$, $(1,10,6,8,4,12)(2,9,5,7,3,11)(13,17)(14,18)(19,20)(21,23)(22,24)(25,35,29,32,27,33)(26,36,30,31,28,34)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$8$:  $D_{4}$
$12$:  $D_{6}$ x 2
$18$:  $D_{9}$
$24$:  $(C_6\times C_2):C_2$ x 2
$36$:  $S_3^2$, $D_{18}$
$72$:  24T61, 36T24
$108$:  $C_3^2 : D_{6} $, 18T50
$216$:  36T232
$324$:  18T132

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Degree 9: None

Degree 12: $(C_6\times C_2):C_2$

Degree 18: 18T132

Low degree siblings

36T1041

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{36}$ $1$ $1$ $0$ $()$
2A $2^{18}$ $1$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B $2^{18}$ $18$ $2$ $18$ $( 1,26)( 2,25)( 3,29)( 4,30)( 5,27)( 6,28)( 7,34)( 8,33)( 9,36)(10,35)(11,31)(12,32)(13,20)(14,19)(15,24)(16,23)(17,21)(18,22)$
2C $2^{17},1^{2}$ $54$ $2$ $17$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5,18)( 6,17)( 7,12)( 8,11)( 9,10)(19,30)(20,29)(21,27)(22,28)(23,26)(24,25)(31,34)(32,33)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 6, 4)( 2, 5, 3)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)(19,22,23)(20,21,24)(25,27,29)(26,28,30)(31,34,36)(32,33,35)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 6, 4)( 2, 5, 3)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)(19,23,22)(20,24,21)(25,29,27)(26,30,28)(31,36,34)(32,35,33)$
3C $3^{6},1^{18}$ $4$ $3$ $12$ $( 1, 4, 6)( 2, 3, 5)( 7,11, 9)( 8,12,10)(13,15,17)(14,16,18)$
3D $3^{8},1^{12}$ $6$ $3$ $16$ $( 7, 9,11)( 8,10,12)(13,15,17)(14,16,18)(19,22,23)(20,21,24)(31,36,34)(32,35,33)$
3E $3^{8},1^{12}$ $12$ $3$ $16$ $( 1, 6, 4)( 2, 5, 3)( 7,11, 9)( 8,12,10)(19,23,22)(20,24,21)(25,27,29)(26,28,30)$
4A $4^{9}$ $54$ $4$ $27$ $( 1,24, 2,23)( 3,19, 4,20)( 5,22, 6,21)( 7,35, 8,36)( 9,33,10,34)(11,32,12,31)(13,26,14,25)(15,28,16,27)(17,30,18,29)$
6A $6^{6}$ $2$ $6$ $30$ $( 1, 3, 6, 2, 4, 5)( 7,12, 9, 8,11,10)(13,16,17,14,15,18)(19,24,22,20,23,21)(25,30,27,26,29,28)(31,35,34,32,36,33)$
6B $6^{6}$ $2$ $6$ $30$ $( 1, 3, 6, 2, 4, 5)( 7,12, 9, 8,11,10)(13,16,17,14,15,18)(19,21,23,20,22,24)(25,28,29,26,27,30)(31,33,36,32,34,35)$
6C $6^{3},2^{9}$ $4$ $6$ $24$ $( 1, 5, 4, 2, 6, 3)( 7,10,11, 8, 9,12)(13,18,15,14,17,16)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
6D $6^{4},2^{6}$ $6$ $6$ $26$ $( 1, 2)( 3, 4)( 5, 6)( 7,12, 9, 8,11,10)(13,18,15,14,17,16)(19,24,22,20,23,21)(25,26)(27,28)(29,30)(31,33,36,32,34,35)$
6E $6^{4},2^{6}$ $12$ $6$ $26$ $( 1, 3, 6, 2, 4, 5)( 7,10,11, 8, 9,12)(13,14)(15,16)(17,18)(19,21,23,20,22,24)(25,30,27,26,29,28)(31,32)(33,34)(35,36)$
6F1 $6^{6}$ $18$ $6$ $30$ $( 1,30, 6,26, 4,28)( 2,29, 5,25, 3,27)( 7,31, 9,34,11,36)( 8,32,10,33,12,35)(13,24,17,20,15,21)(14,23,18,19,16,22)$
6F-1 $6^{6}$ $18$ $6$ $30$ $( 1,28, 4,26, 6,30)( 2,27, 3,25, 5,29)( 7,36,11,34, 9,31)( 8,35,12,33,10,32)(13,21,15,20,17,24)(14,22,16,19,18,23)$
6G1 $6^{4},2^{5},1^{2}$ $54$ $6$ $25$ $( 1,17, 4,15, 6,13)( 2,18, 3,16, 5,14)( 7,12)( 8,11)( 9,10)(19,28,23,30,22,26)(20,27,24,29,21,25)(31,34)(32,33)$
6G-1 $6^{4},2^{5},1^{2}$ $54$ $6$ $25$ $( 1,13, 6,15, 4,17)( 2,14, 5,16, 3,18)( 7,12)( 8,11)( 9,10)(19,26,22,30,23,28)(20,25,21,29,24,27)(31,34)(32,33)$
9A1 $9^{4}$ $6$ $9$ $32$ $( 1,18,10, 6,16,12, 4,14, 8)( 2,17, 9, 5,15,11, 3,13, 7)(19,35,30,22,32,26,23,33,28)(20,36,29,21,31,25,24,34,27)$
9A2 $9^{4}$ $6$ $9$ $32$ $( 1,10,16, 4, 8,18, 6,12,14)( 2, 9,15, 3, 7,17, 5,11,13)(19,30,32,23,28,35,22,26,33)(20,29,31,24,27,36,21,25,34)$
9A4 $9^{4}$ $6$ $9$ $32$ $( 1,16, 8, 6,14,10, 4,18,12)( 2,15, 7, 5,13, 9, 3,17,11)(19,32,28,22,33,30,23,35,26)(20,31,27,21,34,29,24,36,25)$
9B1 $9^{4}$ $12$ $9$ $32$ $( 1,16,12, 6,14, 8, 4,18,10)( 2,15,11, 5,13, 7, 3,17, 9)(19,35,26,22,32,28,23,33,30)(20,36,25,21,31,27,24,34,29)$
9B2 $9^{4}$ $12$ $9$ $32$ $( 1,12,14, 4,10,16, 6, 8,18)( 2,11,13, 3, 9,15, 5, 7,17)(19,26,32,23,30,35,22,28,33)(20,25,31,24,29,36,21,27,34)$
9B4 $9^{4}$ $12$ $9$ $32$ $( 1,14,10, 6,18,12, 4,16, 8)( 2,13, 9, 5,17,11, 3,15, 7)(19,32,30,22,33,26,23,35,28)(20,31,29,21,34,25,24,36,27)$
12A1 $12^{3}$ $54$ $12$ $33$ $( 1,22, 3,24, 6,19, 2,21, 4,23, 5,20)( 7,34,12,35, 9,31, 8,33,11,36,10,32)(13,29,16,26,17,27,14,30,15,25,18,28)$
12A5 $12^{3}$ $54$ $12$ $33$ $( 1,19, 5,24, 4,22, 2,20, 6,23, 3,21)( 7,31,10,35,11,34, 8,32, 9,36,12,33)(13,27,18,26,15,29,14,28,17,25,16,30)$
18A1 $18^{2}$ $6$ $18$ $34$ $( 1,11,18, 3,10,13, 6, 7,16, 2,12,17, 4, 9,14, 5, 8,15)(19,25,35,24,30,34,22,27,32,20,26,36,23,29,33,21,28,31)$
18A5 $18^{2}$ $6$ $18$ $34$ $( 1, 9,16, 3, 8,17, 6,11,14, 2,10,15, 4, 7,18, 5,12,13)(19,29,32,24,28,36,22,25,33,20,30,31,23,27,35,21,26,34)$
18A7 $18^{2}$ $6$ $18$ $34$ $( 1,15, 8, 5,14, 9, 4,17,12, 2,16, 7, 6,13,10, 3,18,11)(19,34,26,21,35,27,23,31,30,20,33,25,22,36,28,24,32,29)$
18B1 $18^{2}$ $12$ $18$ $34$ $( 1, 7,16, 3,12,17, 6, 9,14, 2, 8,15, 4,11,18, 5,10,13)(19,27,35,24,26,34,22,29,32,20,28,36,23,25,33,21,30,31)$
18B5 $18^{2}$ $12$ $18$ $34$ $( 1,11,14, 3,10,15, 6, 7,18, 2,12,13, 4, 9,16, 5, 8,17)(19,25,32,24,30,36,22,27,33,20,26,31,23,29,35,21,28,34)$
18B7 $18^{2}$ $12$ $18$ $34$ $( 1,13, 8, 5,18, 9, 4,15,12, 2,14, 7, 6,17,10, 3,16,11)(19,36,28,21,32,29,23,34,26,20,35,27,22,31,30,24,33,25)$
18C1 $18^{2}$ $18$ $18$ $34$ $( 1,33,14,30,12,23, 6,35,18,26, 8,19, 4,32,16,28,10,22)( 2,34,13,29,11,24, 5,36,17,25, 7,20, 3,31,15,27, 9,21)$
18C-1 $18^{2}$ $18$ $18$ $34$ $( 1,36,14,25,12,20, 6,31,18,27, 8,21, 4,34,16,29,10,24)( 2,35,13,26,11,19, 5,32,17,28, 7,22, 3,33,15,30, 9,23)$
18C5 $18^{2}$ $18$ $18$ $34$ $( 1,23, 8,28,14,35, 4,22,12,26,16,33, 6,19,10,30,18,32)( 2,24, 7,27,13,36, 3,21,11,25,15,34, 5,20, 9,29,17,31)$
18C-5 $18^{2}$ $18$ $18$ $34$ $( 1,32,18,30,10,19, 6,33,16,26,12,22, 4,35,14,28, 8,23)( 2,31,17,29, 9,20, 5,34,15,25,11,21, 3,36,13,27, 7,24)$
18C7 $18^{2}$ $18$ $18$ $34$ $( 1,35,16,30, 8,22, 6,32,14,26,10,23, 4,33,18,28,12,19)( 2,36,15,29, 7,21, 5,31,13,25, 9,24, 3,34,17,27,11,20)$
18C-7 $18^{2}$ $18$ $18$ $34$ $( 1,19,12,28,18,33, 4,23,10,26,14,32, 6,22, 8,30,16,35)( 2,20,11,27,17,34, 3,24, 9,25,13,31, 5,21, 7,29,15,36)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

39 x 39 character table

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Regular extensions

Data not computed