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Properties

Label 36T806
Degree $36$
Order $576$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\GL(2,\mathbb{Z}/4):S_3$

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Copy content magma:G := TransitiveGroup(36, 806);
 

Group invariants

Abstract group:  $\GL(2,\mathbb{Z}/4):S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $576=2^{6} \cdot 3^{2}$
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 $t$:  $806$
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:  $(1,6,2,5)(3,8,4,7)(9,24)(10,23)(11,21)(12,22)(13,32,14,31)(15,29,16,30)(17,25,18,26)(19,27,20,28)(33,36)(34,35)$, $(5,33)(6,34)(7,36)(8,35)(9,30)(10,29)(11,32)(12,31)(13,26)(14,25)(15,28)(16,27)(17,23)(18,24)(19,21)(20,22)$, $(1,11,25,33,13,23,2,12,26,34,14,24)(3,9,27,35,15,22,4,10,28,36,16,21)(5,20,32,8,18,30,6,19,31,7,17,29)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$ x 2
$8$:  $D_{4}$ x 2, $C_2^3$
$12$:  $D_{6}$ x 6
$16$:  $D_4\times C_2$
$24$:  $S_4$, $S_3 \times C_2^2$ x 2
$36$:  $S_3^2$
$48$:  $S_4\times C_2$ x 3, 12T28 x 2
$72$:  12T37
$96$:  12T48
$144$:  12T81, 12T83
$192$:  12T86
$288$:  18T111

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 2

Degree 4: None

Degree 6: $D_{6}$ x 2

Degree 9: $S_3^2$

Degree 12: 12T86

Degree 18: 18T29

Low degree siblings

36T761 x 2, 36T806

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 $2^{18}$ $2$ $2$ $18$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,35)(34,36)$
2C $2^{12},1^{12}$ $3$ $2$ $12$ $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)(21,22)(23,24)(25,26)(27,28)(33,34)(35,36)$
2D $2^{6},1^{24}$ $3$ $2$ $6$ $( 9,10)(11,12)(21,22)(23,24)(33,34)(35,36)$
2E $2^{18}$ $6$ $2$ $18$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)$
2F $2^{15},1^{6}$ $6$ $2$ $15$ $( 1,25)( 2,26)( 3,28)( 4,27)( 5,31)( 6,32)( 7,29)( 8,30)( 9,10)(13,14)(19,20)(21,36)(22,35)(23,34)(24,33)$
2G $2^{15},1^{6}$ $12$ $2$ $15$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)(11,12)(13,19)(14,20)(15,18)(16,17)(23,24)(25,30)(26,29)(27,32)(28,31)(33,34)$
2H $2^{18}$ $18$ $2$ $18$ $( 1,36)( 2,35)( 3,33)( 4,34)( 5, 6)( 7, 8)( 9,26)(10,25)(11,27)(12,28)(13,22)(14,21)(15,24)(16,23)(17,31)(18,32)(19,30)(20,29)$
2I $2^{15},1^{6}$ $18$ $2$ $15$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,36)(10,35)(11,33)(12,34)(23,24)(25,26)(29,30)$
2J $2^{16},1^{4}$ $18$ $2$ $16$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,22)(10,21)(11,23)(12,24)(13,29)(14,30)(15,32)(16,31)(17,28)(18,27)(19,26)(20,25)$
2K $2^{18}$ $36$ $2$ $18$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,13)( 6,14)( 7,15)( 8,16)( 9,34)(10,33)(11,36)(12,35)(21,24)(22,23)(25,32)(26,31)(27,29)(28,30)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,13,26)( 2,14,25)( 3,15,28)( 4,16,27)( 5,18,31)( 6,17,32)( 7,20,30)( 8,19,29)( 9,22,36)(10,21,35)(11,23,34)(12,24,33)$
3B $3^{12}$ $8$ $3$ $24$ $( 1,35, 8)( 2,36, 7)( 3,34, 6)( 4,33, 5)( 9,20,14)(10,19,13)(11,17,15)(12,18,16)(21,29,26)(22,30,25)(23,32,28)(24,31,27)$
3C $3^{12}$ $16$ $3$ $24$ $( 1,19,21)( 2,20,22)( 3,17,23)( 4,18,24)( 5,12,27)( 6,11,28)( 7, 9,25)( 8,10,26)(13,29,35)(14,30,36)(15,32,34)(16,31,33)$
4A $4^{9}$ $6$ $4$ $27$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,24,10,23)(11,22,12,21)(13,28,14,27)(15,25,16,26)(17,30,18,29)(19,32,20,31)(33,35,34,36)$
4B $4^{6},2^{3},1^{6}$ $12$ $4$ $21$ $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,10)(13,19,14,20)(15,18,16,17)(21,22)(25,30,26,29)(27,32,28,31)(35,36)$
4C $4^{9}$ $12$ $4$ $27$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,12,10,11)(13,17,14,18)(15,20,16,19)(21,23,22,24)(25,31,26,32)(27,29,28,30)(33,35,34,36)$
4D $4^{3},2^{12}$ $12$ $4$ $21$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,11,10,12)(13,17)(14,18)(15,20)(16,19)(21,24,22,23)(25,31)(26,32)(27,29)(28,30)(33,36,34,35)$
4E $4^{9}$ $18$ $4$ $27$ $( 1,16, 2,15)( 3,13, 4,14)( 5,20, 6,19)( 7,17, 8,18)( 9,33,10,34)(11,36,12,35)(21,23,22,24)(25,28,26,27)(29,31,30,32)$
4F $4^{6},2^{4},1^{4}$ $18$ $4$ $22$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,18)( 6,17)( 7,20)( 8,19)(13,36,14,35)(15,33,16,34)(21,26,22,25)(23,28,24,27)$
4G $4^{6},2^{6}$ $18$ $4$ $24$ $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,21)(10,22)(11,24)(12,23)(13,30,14,29)(15,31,16,32)(17,28,18,27)(19,26,20,25)(33,34)(35,36)$
4H $4^{6},2^{6}$ $36$ $4$ $24$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5, 9, 6,10)( 7,12, 8,11)(17,35,18,36)(19,34,20,33)(21,31,22,32)(23,30,24,29)(25,28)(26,27)$
6A $6^{6}$ $2$ $6$ $30$ $( 1,14,26, 2,13,25)( 3,16,28, 4,15,27)( 5,17,31, 6,18,32)( 7,19,30, 8,20,29)( 9,21,36,10,22,35)(11,24,34,12,23,33)$
6B $6^{6}$ $4$ $6$ $30$ $( 1,27,13, 4,26,16)( 2,28,14, 3,25,15)( 5,29,18, 8,31,19)( 6,30,17, 7,32,20)( 9,34,22,11,36,23)(10,33,21,12,35,24)$
6C $6^{4},3^{4}$ $6$ $6$ $28$ $( 1,14,26, 2,13,25)( 3,16,28, 4,15,27)( 5,17,31, 6,18,32)( 7,19,30, 8,20,29)( 9,22,36)(10,21,35)(11,23,34)(12,24,33)$
6D $6^{2},3^{8}$ $6$ $6$ $26$ $( 1,13,26)( 2,14,25)( 3,15,28)( 4,16,27)( 5,18,31)( 6,17,32)( 7,20,30)( 8,19,29)( 9,21,36,10,22,35)(11,24,34,12,23,33)$
6E $6^{6}$ $8$ $6$ $30$ $( 1,35, 8, 2,36, 7)( 3,34, 6, 4,33, 5)( 9,20,13,10,19,14)(11,17,16,12,18,15)(21,29,25,22,30,26)(23,32,27,24,31,28)$
6F $6^{6}$ $12$ $6$ $30$ $( 1,15,26, 3,13,28)( 2,16,25, 4,14,27)( 5,20,31, 7,18,30)( 6,19,32, 8,17,29)( 9,23,36,11,22,34)(10,24,35,12,21,33)$
6G $6^{6}$ $16$ $6$ $30$ $( 1,22,19, 2,21,20)( 3,24,17, 4,23,18)( 5,28,12, 6,27,11)( 7,26, 9, 8,25,10)(13,36,29,14,35,30)(15,33,32,16,34,31)$
6H $6^{6}$ $16$ $6$ $30$ $( 1, 5,35, 4, 8,33)( 2, 6,36, 3, 7,34)( 9,15,20,11,14,17)(10,16,19,12,13,18)(21,27,29,24,26,31)(22,28,30,23,25,32)$
6I $6^{6}$ $16$ $6$ $30$ $( 1,31,10, 4,29,12)( 2,32, 9, 3,30,11)( 5,21,16, 8,24,13)( 6,22,15, 7,23,14)(17,36,28,20,34,25)(18,35,27,19,33,26)$
6J $6^{6}$ $16$ $6$ $30$ $( 1,17,22, 4,20,23)( 2,18,21, 3,19,24)( 5,10,28, 8,12,25)( 6, 9,27, 7,11,26)(13,32,36,16,30,34)(14,31,35,15,29,33)$
6K $6^{5},3^{2}$ $24$ $6$ $29$ $( 1,29,13, 8,26,19)( 2,30,14, 7,25,20)( 3,31,15, 5,28,18)( 4,32,16, 6,27,17)( 9,36,22)(10,35,21)(11,33,23,12,34,24)$
6L $6^{5},3^{2}$ $48$ $6$ $29$ $( 1,30,35,25, 8,22)( 2,29,36,26, 7,21)( 3,32,34,28, 6,23)( 4,31,33,27, 5,24)( 9,13,20,10,14,19)(11,15,17)(12,16,18)$
12A $12^{2},6,3^{2}$ $24$ $12$ $31$ $( 1,30,14, 8,26,20, 2,29,13, 7,25,19)( 3,32,16, 5,28,17, 4,31,15, 6,27,18)( 9,35,22,10,36,21)(11,34,23)(12,33,24)$
12B $12^{3}$ $24$ $12$ $33$ $( 1,31,14, 6,26,18, 2,32,13, 5,25,17)( 3,29,16, 7,28,19, 4,30,15, 8,27,20)( 9,34,21,12,36,23,10,33,22,11,35,24)$
12C $12,6^{4}$ $24$ $12$ $31$ $( 1,32,13, 6,26,17)( 2,31,14, 5,25,18)( 3,30,15, 7,28,20)( 4,29,16, 8,27,19)( 9,33,21,11,36,24,10,34,22,12,35,23)$
12D $12^{3}$ $48$ $12$ $33$ $( 1, 5,35, 3, 8,34, 2, 6,36, 4, 7,33)( 9,27,20,24,13,31,10,28,19,23,14,32)(11,25,17,22,16,30,12,26,18,21,15,29)$

Malle's constant $a(G)$:     $1/6$

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Character table

39 x 39 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed