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Properties

Label 36T4787
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, 4787);
 

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 $t$:  $4787$
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:  $(1,16,7,24,4,17,9,22,6,14,11,19)(2,15,8,23,3,18,10,21,5,13,12,20)(25,30,28,26,29,27)(31,32)(33,34)(35,36)$, $(1,27,4,26,6,30)(2,28,3,25,5,29)(7,31,11,35,9,34)(8,32,12,36,10,33)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)$
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: $D_{6}$, $S_4$, $S_4\times C_2$

Degree 9: None

Degree 12: $C_2 \times S_4$

Degree 18: 18T445

Low degree siblings

18T445, 24T7319, 27T538, 36T4786, 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 $2^{12},1^{12}$ $27$ $2$ $12$ $(13,23)(14,24)(15,21)(16,22)(17,19)(18,20)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)$
2B $2^{18}$ $27$ $2$ $18$ $( 1,10)( 2, 9)( 3, 7)( 4, 8)( 5,11)( 6,12)(13,24)(14,23)(15,19)(16,20)(17,21)(18,22)(25,27)(26,28)(29,30)(31,36)(32,35)(33,34)$
2C $2^{18}$ $54$ $2$ $18$ $( 1,31)( 2,32)( 3,33)( 4,34)( 5,36)( 6,35)( 7,30)( 8,29)( 9,27)(10,28)(11,26)(12,25)(13,23)(14,24)(15,20)(16,19)(17,22)(18,21)$
2D $2^{18}$ $54$ $2$ $18$ $( 1,29)( 2,30)( 3,26)( 4,25)( 5,27)( 6,28)( 7,36)( 8,35)( 9,33)(10,34)(11,32)(12,31)(13,22)(14,21)(15,19)(16,20)(17,23)(18,24)$
2E $2^{18}$ $81$ $2$ $18$ $( 1, 5)( 2, 6)( 3, 4)( 7, 8)( 9,12)(10,11)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,28)(29,30)(31,36)(32,35)(33,34)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 4, 6)( 2, 3, 5)( 7, 9,11)( 8,10,12)(13,15,18)(14,16,17)(19,24,22)(20,23,21)(25,28,29)(26,27,30)(31,35,34)(32,36,33)$
3B $3^{4},1^{24}$ $6$ $3$ $8$ $( 1, 6, 4)( 2, 5, 3)( 7, 9,11)( 8,10,12)$
3C $3^{12}$ $8$ $3$ $24$ $( 1, 4, 6)( 2, 3, 5)( 7,11, 9)( 8,12,10)(13,15,18)(14,16,17)(19,22,24)(20,21,23)(25,28,29)(26,27,30)(31,34,35)(32,33,36)$
3D $3^{8},1^{12}$ $12$ $3$ $16$ $( 1, 6, 4)( 2, 5, 3)( 7, 9,11)( 8,10,12)(13,15,18)(14,16,17)(19,22,24)(20,21,23)$
3E $3^{10},1^{6}$ $12$ $3$ $20$ $( 1, 6, 4)( 2, 5, 3)( 7,11, 9)( 8,12,10)(19,24,22)(20,23,21)(25,29,28)(26,30,27)(31,34,35)(32,33,36)$
3F $3^{6},1^{18}$ $16$ $3$ $12$ $( 1, 6, 4)( 2, 5, 3)(13,18,15)(14,17,16)(25,29,28)(26,30,27)$
3G $3^{8},1^{12}$ $24$ $3$ $16$ $( 7,11, 9)( 8,12,10)(13,18,15)(14,17,16)(25,28,29)(26,27,30)(31,35,34)(32,36,33)$
3H $3^{12}$ $72$ $3$ $24$ $( 1,27,23)( 2,28,24)( 3,29,22)( 4,30,21)( 5,25,19)( 6,26,20)( 7,34,18)( 8,33,17)( 9,31,13)(10,32,14)(11,35,15)(12,36,16)$
3I $3^{12}$ $144$ $3$ $24$ $( 1,21,30)( 2,22,29)( 3,19,25)( 4,20,26)( 5,24,28)( 6,23,27)( 7,18,31)( 8,17,32)( 9,13,35)(10,14,36)(11,15,34)(12,16,33)$
4A $4^{6},2^{6}$ $54$ $4$ $24$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,36,23,25)(14,35,24,26)(15,33,21,28)(16,34,22,27)(17,31,19,30)(18,32,20,29)$
4B $4^{6},2^{4},1^{4}$ $486$ $4$ $22$ $( 3, 5)( 4, 6)( 7, 9)( 8,10)(13,35,23,30)(14,36,24,29)(15,31,21,27)(16,32,22,28)(17,33,19,25)(18,34,20,26)$
6A $3^{4},2^{12}$ $54$ $6$ $20$ $( 1, 4, 6)( 2, 3, 5)( 7,11, 9)( 8,12,10)(13,23)(14,24)(15,21)(16,22)(17,19)(18,20)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)$
6B $6^{4},3^{4}$ $54$ $6$ $28$ $( 1, 9, 4,11, 6, 7)( 2,10, 3,12, 5, 8)(13,18,15)(14,17,16)(19,22,24)(20,21,23)(25,33,28,32,29,36)(26,34,27,31,30,35)$
6C $6^{6}$ $108$ $6$ $30$ $( 1,28, 4,29, 6,25)( 2,27, 3,30, 5,26)( 7,32, 9,36,11,33)( 8,31,10,35,12,34)(13,24,15,22,18,19)(14,23,16,21,17,20)$
6D $6^{2},2^{12}$ $108$ $6$ $22$ $( 1,13)( 2,14)( 3,17)( 4,18)( 5,16)( 6,15)( 7,23)( 8,24)( 9,20)(10,19)(11,21)(12,22)(25,36,29,33,28,32)(26,35,30,34,27,31)$
6E $6^{4},2^{6}$ $108$ $6$ $26$ $( 1,19, 6,22, 4,24)( 2,20, 5,21, 3,23)( 7,14, 9,16,11,17)( 8,13,10,15,12,18)(25,35)(26,36)(27,33)(28,34)(29,31)(30,32)$
6F $6^{4},3^{2},1^{6}$ $108$ $6$ $24$ $( 1, 9, 6, 7, 4,11)( 2,10, 5, 8, 3,12)(19,22,24)(20,21,23)(25,33,29,36,28,32)(26,34,30,35,27,31)$
6G $6^{4},2^{6}$ $108$ $6$ $26$ $( 1,18, 4,15, 6,13)( 2,17, 3,16, 5,14)( 7,20,11,23, 9,21)( 8,19,12,24,10,22)(25,36)(26,35)(27,31)(28,32)(29,33)(30,34)$
6H $6^{4},2^{6}$ $108$ $6$ $26$ $( 1, 8, 6,10, 4,12)( 2, 7, 5, 9, 3,11)(13,19,18,24,15,22)(14,20,17,23,16,21)(25,27)(26,28)(29,30)(31,36)(32,35)(33,34)$
6I $6^{2},2^{12}$ $108$ $6$ $22$ $( 1, 3)( 2, 4)( 5, 6)( 7,10)( 8, 9)(11,12)(13,22,15,24,18,19)(14,21,16,23,17,20)(25,35)(26,36)(27,32)(28,31)(29,34)(30,33)$
6J $6^{6}$ $216$ $6$ $30$ $( 1,35, 4,31, 6,34)( 2,36, 3,32, 5,33)( 7,27,11,30, 9,26)( 8,28,12,29,10,25)(13,21,15,23,18,20)(14,22,16,24,17,19)$
6K $6^{4},2^{6}$ $216$ $6$ $26$ $( 1,22)( 2,21)( 3,20)( 4,19)( 5,23)( 6,24)( 7,17,11,16, 9,14)( 8,18,12,15,10,13)(25,31,28,35,29,34)(26,32,27,36,30,33)$
6L $6^{6}$ $648$ $6$ $30$ $( 1,19,27, 5,23,25)( 2,20,28, 6,24,26)( 3,21,29, 4,22,30)( 7,17,34, 8,18,33)( 9,16,31,12,13,36)(10,15,32,11,14,35)$
9A $9^{4}$ $144$ $9$ $32$ $( 1,23,30, 4,21,26, 6,20,27)( 2,24,29, 3,22,25, 5,19,28)( 7,18,35,11,15,31, 9,13,34)( 8,17,36,12,16,32,10,14,33)$
9B1 $9^{4}$ $144$ $9$ $32$ $( 1,30,20, 6,27,21, 4,26,23)( 2,29,19, 5,28,22, 3,25,24)( 7,34,18, 9,31,13,11,35,15)( 8,33,17,10,32,14,12,36,16)$
9B-1 $9^{4}$ $144$ $9$ $32$ $( 1,23,26, 4,21,27, 6,20,30)( 2,24,25, 3,22,28, 5,19,29)( 7,15,35,11,13,31, 9,18,34)( 8,16,36,12,14,32,10,17,33)$
12A $6^{2},4^{6}$ $108$ $12$ $28$ $( 1, 5, 4, 2, 6, 3)( 7,10,11, 8, 9,12)(13,25,23,36)(14,26,24,35)(15,28,21,33)(16,27,22,34)(17,30,19,31)(18,29,20,32)$
12B $12^{2},6^{2}$ $108$ $12$ $32$ $( 1,33, 9,28, 4,32,11,29, 6,36, 7,25)( 2,34,10,27, 3,31,12,30, 5,35, 8,26)(13,16,18,14,15,17)(19,23,22,20,24,21)$
12C1 $12^{2},6,2^{3}$ $108$ $12$ $30$ $( 1,36, 9,28, 6,32, 7,25, 4,33,11,29)( 2,35,10,27, 5,31, 8,26, 3,34,12,30)(13,14)(15,16)(17,18)(19,23,22,20,24,21)$
12C-1 $12^{2},6,2^{3}$ $108$ $12$ $30$ $( 1,29,11,33, 4,25, 7,32, 6,28, 9,36)( 2,30,12,34, 3,26, 8,31, 5,27,10,35)(13,14)(15,16)(17,18)(19,21,24,20,22,23)$

Malle's constant $a(G)$:     $1/8$

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Character table

36 x 36 character table

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Regular extensions

Data not computed