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Properties

Label 36T4786
Degree $36$
Order $3888$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:(S_3\times S_4)$

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Copy content magma:G := TransitiveGroup(36, 4786);
 

Group invariants

Abstract group:  $C_3^3:(S_3\times S_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $3888=2^{4} \cdot 3^{5}$
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 $t$:  $4786$
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:  $(1,30,6,27,4,26)(2,29,5,28,3,25)(7,31,11,34,9,35)(8,32,12,33,10,36)(13,22,17,23,15,19)(14,21,18,24,16,20)$, $(1,5)(2,6)(3,4)(7,8)(9,12)(10,11)(13,33,21,28)(14,34,22,27)(15,36,20,25)(16,35,19,26)(17,32,24,29)(18,31,23,30)$
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 2
$12$:  $D_{6}$ x 2
$24$:  $S_4$
$36$:  $S_3^2$
$48$:  $S_4\times C_2$
$144$:  12T83
$1296$:  $S_3\wr S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$, $S_4$, $S_4$

Degree 9: None

Degree 12: $S_4$

Degree 18: 18T445

Low degree siblings

18T445, 24T7319, 27T538, 36T4787, 36T4788, 36T4820, 36T4951, 36T4974 x 2

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 $2^{12},1^{12}$ $27$ $2$ $12$ $( 1,11)( 2,12)( 3, 8)( 4, 7)( 5,10)( 6, 9)(13,21)(14,22)(15,20)(16,19)(17,24)(18,23)$
2B $2^{16},1^{4}$ $27$ $2$ $16$ $( 1, 9)( 2,10)( 3, 8)( 4, 7)( 5,12)( 6,11)(13,15)(14,16)(19,23)(20,24)(25,36)(26,35)(27,31)(28,32)(29,33)(30,34)$
2C $2^{18}$ $54$ $2$ $18$ $( 1,10)( 2, 9)( 3, 7)( 4, 8)( 5,11)( 6,12)(13,36)(14,35)(15,32)(16,31)(17,33)(18,34)(19,27)(20,28)(21,29)(22,30)(23,26)(24,25)$
2D $2^{18}$ $54$ $2$ $18$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5, 7)( 6, 8)(13,28)(14,27)(15,29)(16,30)(17,25)(18,26)(19,34)(20,33)(21,36)(22,35)(23,31)(24,32)$
2E $2^{12},1^{12}$ $81$ $2$ $12$ $( 1, 4)( 2, 3)( 7, 9)( 8,10)(13,15)(14,16)(19,23)(20,24)(25,28)(26,27)(31,35)(32,36)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 6, 4)( 2, 5, 3)( 7,11, 9)( 8,12,10)(13,17,15)(14,18,16)(19,22,23)(20,21,24)(25,29,28)(26,30,27)(31,34,35)(32,33,36)$
3B $3^{4},1^{24}$ $6$ $3$ $8$ $(25,28,29)(26,27,30)(31,34,35)(32,33,36)$
3C $3^{12}$ $8$ $3$ $24$ $( 1, 6, 4)( 2, 5, 3)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)(19,23,22)(20,24,21)(25,29,28)(26,30,27)(31,35,34)(32,36,33)$
3D $3^{8},1^{12}$ $12$ $3$ $16$ $( 1, 6, 4)( 2, 5, 3)( 7, 9,11)( 8,10,12)(25,28,29)(26,27,30)(31,34,35)(32,33,36)$
3E $3^{10},1^{6}$ $12$ $3$ $20$ $( 7,11, 9)( 8,12,10)(13,15,17)(14,16,18)(19,23,22)(20,24,21)(25,28,29)(26,27,30)(31,35,34)(32,36,33)$
3F $3^{6},1^{18}$ $16$ $3$ $12$ $( 1, 4, 6)( 2, 3, 5)(13,15,17)(14,16,18)(25,28,29)(26,27,30)$
3G $3^{8},1^{12}$ $24$ $3$ $16$ $( 1, 4, 6)( 2, 3, 5)(13,17,15)(14,18,16)(19,22,23)(20,21,24)(31,35,34)(32,36,33)$
3H $3^{12}$ $72$ $3$ $24$ $( 1,32,19)( 2,31,20)( 3,35,24)( 4,36,23)( 5,34,21)( 6,33,22)( 7,25,14)( 8,26,13)( 9,28,16)(10,27,15)(11,29,18)(12,30,17)$
3I $3^{12}$ $144$ $3$ $24$ $( 1,22,32)( 2,21,31)( 3,20,35)( 4,19,36)( 5,24,34)( 6,23,33)( 7,14,28)( 8,13,27)( 9,16,29)(10,15,30)(11,18,25)(12,17,26)$
4A $4^{6},2^{6}$ $54$ $4$ $24$ $( 1,21,11,13)( 2,22,12,14)( 3,19, 8,16)( 4,20, 7,15)( 5,23,10,18)( 6,24, 9,17)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
4B $4^{6},2^{6}$ $486$ $4$ $24$ $( 1,24,11,13)( 2,23,12,14)( 3,19, 8,18)( 4,20, 7,17)( 5,22,10,16)( 6,21, 9,15)(25,26)(27,29)(28,30)(31,32)(33,35)(34,36)$
6A $3^{4},2^{12}$ $54$ $6$ $20$ $( 1,11)( 2,12)( 3, 8)( 4, 7)( 5,10)( 6, 9)(13,21)(14,22)(15,20)(16,19)(17,24)(18,23)(25,29,28)(26,30,27)(31,35,34)(32,36,33)$
6B $6^{4},3^{4}$ $54$ $6$ $28$ $( 1, 4, 6)( 2, 3, 5)( 7, 9,11)( 8,10,12)(13,24,17,20,15,21)(14,23,18,19,16,22)(25,36,29,32,28,33)(26,35,30,31,27,34)$
6C $6^{6}$ $108$ $6$ $30$ $( 1,12, 6,10, 4, 8)( 2,11, 5, 9, 3, 7)(13,29,17,28,15,25)(14,30,18,27,16,26)(19,31,22,34,23,35)(20,32,21,33,24,36)$
6D $6^{2},2^{12}$ $108$ $6$ $22$ $( 1,27)( 2,28)( 3,25)( 4,26)( 5,29)( 6,30)( 7,34)( 8,33)( 9,35)(10,36)(11,31)(12,32)(13,23,15,19,17,22)(14,24,16,20,18,21)$
6E $6^{4},2^{6}$ $108$ $6$ $26$ $( 1,31, 6,34, 4,35)( 2,32, 5,33, 3,36)( 7,26, 9,27,11,30)( 8,25,10,28,12,29)(13,22)(14,21)(15,19)(16,20)(17,23)(18,24)$
6F $6^{4},3^{2},1^{6}$ $108$ $6$ $24$ $( 7, 9,11)( 8,10,12)(13,24,15,21,17,20)(14,23,16,22,18,19)(25,36,28,33,29,32)(26,35,27,34,30,31)$
6G $6^{4},2^{6}$ $108$ $6$ $26$ $( 1,27, 4,26, 6,30)( 2,28, 3,25, 5,29)( 7,35,11,34, 9,31)( 8,36,12,33,10,32)(13,23)(14,24)(15,19)(16,20)(17,22)(18,21)$
6H $6^{4},2^{4},1^{4}$ $108$ $6$ $24$ $( 1,11, 4, 9, 6, 7)( 2,12, 3,10, 5, 8)(13,15)(14,16)(19,23)(20,24)(25,33,28,36,29,32)(26,34,27,35,30,31)$
6I $6^{2},2^{10},1^{4}$ $108$ $6$ $20$ $( 1, 7, 6, 9, 4,11)( 2, 8, 5,10, 3,12)(13,24)(14,23)(15,20)(16,19)(17,21)(18,22)(25,29)(26,30)(33,36)(34,35)$
6J $6^{6}$ $216$ $6$ $30$ $( 1, 8, 6,10, 4,12)( 2, 7, 5, 9, 3,11)(13,32,17,36,15,33)(14,31,18,35,16,34)(19,30,23,27,22,26)(20,29,24,28,21,25)$
6K $6^{4},2^{6}$ $216$ $6$ $26$ $( 1,35, 4,34, 6,31)( 2,36, 3,33, 5,32)( 7,27)( 8,28)( 9,30)(10,29)(11,26)(12,25)(13,19,17,22,15,23)(14,20,18,21,16,24)$
6L $6^{4},3^{4}$ $648$ $6$ $28$ $( 1,23,32, 4,19,36)( 2,24,31, 3,20,35)( 5,21,34)( 6,22,33)( 7,16,25, 9,14,28)( 8,15,26,10,13,27)(11,18,29)(12,17,30)$
9A $9^{4}$ $144$ $9$ $32$ $( 1,23,36, 4,22,33, 6,19,32)( 2,24,35, 3,21,34, 5,20,31)( 7,18,29,11,16,28, 9,14,25)( 8,17,30,12,15,27,10,13,26)$
9B1 $9^{4}$ $144$ $9$ $32$ $( 1,33,22, 6,36,23, 4,32,19)( 2,34,21, 5,35,24, 3,31,20)( 7,25,18, 9,28,14,11,29,16)( 8,26,17,10,27,13,12,30,15)$
9B-1 $9^{4}$ $144$ $9$ $32$ $( 1,19,32, 4,23,36, 6,22,33)( 2,20,31, 3,24,35, 5,21,34)( 7,16,29,11,14,28, 9,18,25)( 8,15,30,12,13,27,10,17,26)$
12A $6^{2},4^{6}$ $108$ $12$ $28$ $( 1,13,11,21)( 2,14,12,22)( 3,16, 8,19)( 4,15, 7,20)( 5,18,10,23)( 6,17, 9,24)(25,27,29,26,28,30)(31,33,35,32,34,36)$
12B $12^{2},6^{2}$ $108$ $12$ $32$ $( 1, 5, 4, 2, 6, 3)( 7,12, 9, 8,11,10)(13,29,24,32,17,28,20,33,15,25,21,36)(14,30,23,31,18,27,19,34,16,26,22,35)$
12C1 $12^{2},6,2^{3}$ $108$ $12$ $30$ $( 1, 2)( 3, 4)( 5, 6)( 7,12, 9, 8,11,10)(13,28,24,33,15,29,21,32,17,25,20,36)(14,27,23,34,16,30,22,31,18,26,19,35)$
12C-1 $12^{2},6,2^{3}$ $108$ $12$ $30$ $( 1, 2)( 3, 4)( 5, 6)( 7,10,11, 8, 9,12)(13,36,20,25,17,32,21,29,15,33,24,28)(14,35,19,26,18,31,22,30,16,34,23,27)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

36 x 36 character table

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Regular extensions

Data not computed