↑

Properties

Label 36T492
Degree $36$
Order $324$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:D_6$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 492);
 

Group invariants

Abstract group:  $C_3^3:D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $324=2^{2} \cdot 3^{4}$
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 $t$:  $492$
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:  $(1,25)(2,26)(3,28)(4,27)(5,29)(6,30)(7,24)(8,23)(9,21)(10,22)(11,19)(12,20)(13,36)(14,35)(15,32)(16,31)(17,34)(18,33)$, $(1,8,17,2,7,18)(3,12,14,4,11,13)(5,9,16,6,10,15)(19,25,31,20,26,32)(21,28,34,22,27,33)(23,30,35,24,29,36)$, $(1,22,6,19,4,23)(2,21,5,20,3,24)(7,33,9,31,12,35)(8,34,10,32,11,36)(13,29,17,28,15,26)(14,30,18,27,16,25)$, $(1,15,7)(2,16,8)(3,18,11)(4,17,12)(5,14,10)(6,13,9)(19,35,26)(20,36,25)(21,32,27)(22,31,28)(23,33,29)(24,34,30)$
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 13
$12$:  $D_{6}$ x 13
$18$:  $C_3^2:C_2$ x 13
$36$:  18T12 x 13
$54$:  $(C_3^2:C_3):C_2$, 27T7
$108$:  18T52
$162$:  18T89

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Degree 9: None

Degree 12: $D_6$

Degree 18: 18T89

Low degree siblings

36T492 x 3

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 $2^{18}$ $1$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B $2^{18}$ $27$ $2$ $18$ $( 1,31)( 2,32)( 3,34)( 4,33)( 5,36)( 6,35)( 7,28)( 8,27)( 9,26)(10,25)(11,30)(12,29)(13,19)(14,20)(15,22)(16,21)(17,23)(18,24)$
2C $2^{18}$ $27$ $2$ $18$ $( 1,25)( 2,26)( 3,28)( 4,27)( 5,29)( 6,30)( 7,20)( 8,19)( 9,24)(10,23)(11,22)(12,21)(13,34)(14,33)(15,36)(16,35)(17,32)(18,31)$
3A1 $3^{12}$ $1$ $3$ $24$ $( 1, 6, 4)( 2, 5, 3)( 7, 9,12)( 8,10,11)(13,17,15)(14,18,16)(19,23,22)(20,24,21)(25,30,27)(26,29,28)(31,35,33)(32,36,34)$
3A-1 $3^{12}$ $1$ $3$ $24$ $( 1, 4, 6)( 2, 3, 5)( 7,12, 9)( 8,11,10)(13,15,17)(14,16,18)(19,22,23)(20,21,24)(25,27,30)(26,28,29)(31,33,35)(32,34,36)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 6, 4)( 2, 5, 3)( 7, 9,12)( 8,10,11)(13,17,15)(14,18,16)(19,22,23)(20,21,24)(25,27,30)(26,28,29)(31,33,35)(32,34,36)$
3C1 $3^{6},1^{18}$ $2$ $3$ $12$ $( 1, 4, 6)( 2, 3, 5)( 7,12, 9)( 8,11,10)(13,15,17)(14,16,18)$
3C-1 $3^{6},1^{18}$ $2$ $3$ $12$ $( 1, 6, 4)( 2, 5, 3)( 7, 9,12)( 8,10,11)(13,17,15)(14,18,16)$
3D $3^{8},1^{12}$ $6$ $3$ $16$ $( 7, 9,12)( 8,10,11)(13,15,17)(14,16,18)(19,22,23)(20,21,24)(31,35,33)(32,36,34)$
3E $3^{12}$ $6$ $3$ $24$ $( 1,12,15)( 2,11,16)( 3,10,18)( 4, 9,17)( 5, 8,14)( 6, 7,13)(19,29,33)(20,30,34)(21,25,36)(22,26,35)(23,28,31)(24,27,32)$
3F $3^{8},1^{12}$ $6$ $3$ $16$ $( 1, 4, 6)( 2, 3, 5)(13,17,15)(14,18,16)(25,30,27)(26,29,28)(31,33,35)(32,34,36)$
3G $3^{12}$ $6$ $3$ $24$ $( 1, 9,13)( 2,10,14)( 3, 8,16)( 4, 7,15)( 5,11,18)( 6,12,17)(19,28,35)(20,27,36)(21,30,32)(22,29,31)(23,26,33)(24,25,34)$
3H $3^{8},1^{12}$ $6$ $3$ $16$ $( 1, 4, 6)( 2, 3, 5)( 7, 9,12)( 8,10,11)(19,22,23)(20,21,24)(25,30,27)(26,29,28)$
3I $3^{12}$ $6$ $3$ $24$ $( 1,17, 7)( 2,18, 8)( 3,14,11)( 4,13,12)( 5,16,10)( 6,15, 9)(19,31,26)(20,32,25)(21,34,27)(22,33,28)(23,35,29)(24,36,30)$
3J $3^{12}$ $6$ $3$ $24$ $( 1, 7,13)( 2, 8,14)( 3,11,16)( 4,12,15)( 5,10,18)( 6, 9,17)(19,26,33)(20,25,34)(21,27,36)(22,28,35)(23,29,31)(24,30,32)$
3K $3^{12}$ $6$ $3$ $24$ $( 1,13,12)( 2,14,11)( 3,16,10)( 4,15, 9)( 5,18, 8)( 6,17, 7)(19,35,26)(20,36,25)(21,32,27)(22,31,28)(23,33,29)(24,34,30)$
3L $3^{12}$ $6$ $3$ $24$ $( 1, 7,15)( 2, 8,16)( 3,11,18)( 4,12,17)( 5,10,14)( 6, 9,13)(19,28,33)(20,27,34)(21,30,36)(22,29,35)(23,26,31)(24,25,32)$
3M $3^{12}$ $6$ $3$ $24$ $( 1, 7,15)( 2, 8,16)( 3,11,18)( 4,12,17)( 5,10,14)( 6, 9,13)(19,26,35)(20,25,36)(21,27,32)(22,28,31)(23,29,33)(24,30,34)$
3N $3^{12}$ $6$ $3$ $24$ $( 1, 7,13)( 2, 8,14)( 3,11,16)( 4,12,15)( 5,10,18)( 6, 9,17)(19,29,35)(20,30,36)(21,25,32)(22,26,31)(23,28,33)(24,27,34)$
3O $3^{12}$ $6$ $3$ $24$ $( 1,15, 9)( 2,16,10)( 3,18, 8)( 4,17, 7)( 5,14,11)( 6,13,12)(19,33,26)(20,34,25)(21,36,27)(22,35,28)(23,31,29)(24,32,30)$
6A1 $6^{6}$ $1$ $6$ $30$ $( 1, 3, 6, 2, 4, 5)( 7,11, 9, 8,12,10)(13,16,17,14,15,18)(19,21,23,20,22,24)(25,28,30,26,27,29)(31,34,35,32,33,36)$
6A-1 $6^{6}$ $1$ $6$ $30$ $( 1, 5, 4, 2, 6, 3)( 7,10,12, 8, 9,11)(13,18,15,14,17,16)(19,24,22,20,23,21)(25,29,27,26,30,28)(31,36,33,32,35,34)$
6B $6^{6}$ $2$ $6$ $30$ $( 1, 3, 6, 2, 4, 5)( 7,11, 9, 8,12,10)(13,16,17,14,15,18)(19,24,22,20,23,21)(25,29,27,26,30,28)(31,36,33,32,35,34)$
6C1 $6^{3},2^{9}$ $2$ $6$ $24$ $( 1, 5, 4, 2, 6, 3)( 7,10,12, 8, 9,11)(13,18,15,14,17,16)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
6C-1 $6^{3},2^{9}$ $2$ $6$ $24$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,21,23,20,22,24)(25,28,30,26,27,29)(31,34,35,32,33,36)$
6D $6^{4},2^{6}$ $6$ $6$ $26$ $( 1, 2)( 3, 4)( 5, 6)( 7,11, 9, 8,12,10)(13,18,15,14,17,16)(19,24,22,20,23,21)(25,26)(27,28)(29,30)(31,34,35,32,33,36)$
6E $6^{6}$ $6$ $6$ $30$ $( 1,16,12, 2,15,11)( 3,17,10, 4,18, 9)( 5,13, 8, 6,14, 7)(19,34,29,20,33,30)(21,35,25,22,36,26)(23,32,28,24,31,27)$
6F $6^{4},2^{6}$ $6$ $6$ $26$ $( 1, 5, 4, 2, 6, 3)( 7, 8)( 9,10)(11,12)(13,16,17,14,15,18)(19,20)(21,22)(23,24)(25,28,30,26,27,29)(31,36,33,32,35,34)$
6G $6^{6}$ $6$ $6$ $30$ $( 1,14, 9, 2,13,10)( 3,15, 8, 4,16, 7)( 5,17,11, 6,18,12)(19,36,28,20,35,27)(21,31,30,22,32,29)(23,34,26,24,33,25)$
6H $6^{4},2^{6}$ $6$ $6$ $26$ $( 1, 5, 4, 2, 6, 3)( 7,11, 9, 8,12,10)(13,14)(15,16)(17,18)(19,24,22,20,23,21)(25,28,30,26,27,29)(31,32)(33,34)(35,36)$
6I $6^{6}$ $6$ $6$ $30$ $( 1, 8,17, 2, 7,18)( 3,12,14, 4,11,13)( 5, 9,16, 6,10,15)(19,25,31,20,26,32)(21,28,34,22,27,33)(23,30,35,24,29,36)$
6J $6^{6}$ $6$ $6$ $30$ $( 1,14, 7, 2,13, 8)( 3,15,11, 4,16,12)( 5,17,10, 6,18, 9)(19,34,26,20,33,25)(21,35,27,22,36,28)(23,32,29,24,31,30)$
6K $6^{6}$ $6$ $6$ $30$ $( 1,11,13, 2,12,14)( 3, 9,16, 4,10,15)( 5, 7,18, 6, 8,17)(19,25,35,20,26,36)(21,28,32,22,27,31)(23,30,33,24,29,34)$
6L $6^{6}$ $6$ $6$ $30$ $( 1,16, 7, 2,15, 8)( 3,17,11, 4,18,12)( 5,13,10, 6,14, 9)(19,34,28,20,33,27)(21,35,30,22,36,29)(23,32,26,24,31,25)$
6M $6^{6}$ $6$ $6$ $30$ $( 1,16, 7, 2,15, 8)( 3,17,11, 4,18,12)( 5,13,10, 6,14, 9)(19,36,26,20,35,25)(21,31,27,22,32,28)(23,34,29,24,33,30)$
6N $6^{6}$ $6$ $6$ $30$ $( 1,14, 7, 2,13, 8)( 3,15,11, 4,16,12)( 5,17,10, 6,18, 9)(19,36,29,20,35,30)(21,31,25,22,32,26)(23,34,28,24,33,27)$
6O $6^{6}$ $6$ $6$ $30$ $( 1,10,15, 2, 9,16)( 3, 7,18, 4, 8,17)( 5,12,14, 6,11,13)(19,25,33,20,26,34)(21,28,36,22,27,35)(23,30,31,24,29,32)$
6P1 $6^{6}$ $27$ $6$ $30$ $( 1,33, 6,31, 4,35)( 2,34, 5,32, 3,36)( 7,29, 9,28,12,26)( 8,30,10,27,11,25)(13,22,17,19,15,23)(14,21,18,20,16,24)$
6P-1 $6^{6}$ $27$ $6$ $30$ $( 1,35, 4,31, 6,33)( 2,36, 3,32, 5,34)( 7,26,12,28, 9,29)( 8,25,11,27,10,30)(13,23,15,19,17,22)(14,24,16,20,18,21)$
6Q1 $6^{6}$ $27$ $6$ $30$ $( 1,27, 6,25, 4,30)( 2,28, 5,26, 3,29)( 7,21, 9,20,12,24)( 8,22,10,19,11,23)(13,36,17,34,15,32)(14,35,18,33,16,31)$
6Q-1 $6^{6}$ $27$ $6$ $30$ $( 1,30, 4,25, 6,27)( 2,29, 3,26, 5,28)( 7,24,12,20, 9,21)( 8,23,11,19,10,22)(13,32,15,34,17,36)(14,31,16,33,18,35)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

42 x 42 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed