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Properties

Label 36T4983
Degree $36$
Order $4320$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $D_6\times A_6$

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Copy content magma:G := TransitiveGroup(36, 4983);
 

Group invariants

Abstract group:  $D_6\times A_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $4320=2^{5} \cdot 3^{3} \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
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Group action invariants

Degree $n$:  $36$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4983$
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:  $(1,16,35,26,4,18,32,27,6,14,33,30)(2,15,36,25,3,17,31,28,5,13,34,29)(7,21,11,20,10,24)(8,22,12,19,9,23)$, $(1,29)(2,30)(3,27)(4,28)(5,26)(6,25)(7,18)(8,17)(9,15)(10,16)(11,14)(12,13)(19,23)(20,24)(31,36)(32,35)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$360$:  $A_6$
$720$:  12T180 x 3
$1440$:  24T2952
$2160$:  18T363

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$, $A_6$

Degree 9: None

Degree 12: 12T180

Degree 18: 18T363

Low degree siblings

36T4983 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 $2^{18}$ $3$ $2$ $18$ $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32)(33,36)(34,35)$
2C $2^{12},1^{12}$ $3$ $2$ $12$ $( 1, 6)( 2, 5)( 7,11)( 8,12)(13,17)(14,18)(19,23)(20,24)(25,29)(26,30)(31,36)(32,35)$
2D $2^{12},1^{12}$ $45$ $2$ $12$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,18)( 6,17)( 7,26)( 8,25)( 9,28)(10,27)(11,30)(12,29)$
2E $2^{18}$ $45$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7,20)( 8,19)( 9,22)(10,21)(11,24)(12,23)(13,14)(15,16)(17,18)(25,31)(26,32)(27,33)(28,34)(29,36)(30,35)$
2F $2^{16},1^{4}$ $135$ $2$ $16$ $( 1,13)( 2,14)( 3,18)( 4,17)( 5,16)( 6,15)( 9,12)(10,11)(19,31)(20,32)(21,35)(22,36)(23,34)(24,33)(27,30)(28,29)$
2G $2^{18}$ $135$ $2$ $18$ $( 1, 3)( 2, 4)( 5, 6)( 7,33)( 8,34)( 9,31)(10,32)(11,35)(12,36)(13,27)(14,28)(15,26)(16,25)(17,30)(18,29)(19,21)(20,22)(23,24)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 4, 6)( 2, 3, 5)( 7,10,11)( 8, 9,12)(13,15,17)(14,16,18)(19,22,23)(20,21,24)(25,28,29)(26,27,30)(31,34,36)(32,33,35)$
3B $3^{12}$ $40$ $3$ $24$ $( 1,32,25)( 2,31,26)( 3,34,27)( 4,33,28)( 5,36,30)( 6,35,29)( 7,14,19)( 8,13,20)( 9,15,21)(10,16,22)(11,18,23)(12,17,24)$
3C $3^{6},1^{18}$ $40$ $3$ $12$ $( 1,25,32)( 2,26,31)( 3,27,34)( 4,28,33)( 5,30,36)( 6,29,35)$
3D $3^{12}$ $80$ $3$ $24$ $( 1,21,29)( 2,22,30)( 3,23,26)( 4,24,25)( 5,19,27)( 6,20,28)( 7,16,36)( 8,15,35)( 9,17,32)(10,18,31)(11,14,34)(12,13,33)$
3E $3^{12}$ $80$ $3$ $24$ $( 1, 4, 6)( 2, 3, 5)( 7,10,11)( 8, 9,12)(13,15,17)(14,16,18)(19,34,30)(20,33,29)(21,35,25)(22,36,26)(23,31,27)(24,32,28)$
4A $4^{6},2^{6}$ $90$ $4$ $24$ $( 1,25,13, 8)( 2,26,14, 7)( 3,27,16,10)( 4,28,15, 9)( 5,30,18,11)( 6,29,17,12)(19,31)(20,32)(21,33)(22,34)(23,36)(24,35)$
4B $4^{6},2^{6}$ $90$ $4$ $24$ $( 1,14,32, 7)( 2,13,31, 8)( 3,15,34, 9)( 4,16,33,10)( 5,17,36,12)( 6,18,35,11)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)$
4C $4^{6},2^{6}$ $270$ $4$ $24$ $( 1,22, 8,27)( 2,21, 7,28)( 3,20,10,25)( 4,19, 9,26)( 5,24,11,29)( 6,23,12,30)(13,34)(14,33)(15,31)(16,32)(17,36)(18,35)$
4D $4^{6},2^{6}$ $270$ $4$ $24$ $( 1,21,13, 9)( 2,22,14,10)( 3,19,16, 7)( 4,20,15, 8)( 5,23,18,11)( 6,24,17,12)(25,33)(26,34)(27,31)(28,32)(29,35)(30,36)$
5A1 $5^{6},1^{6}$ $72$ $5$ $24$ $( 1, 8,25,20,32)( 2, 7,26,19,31)( 3,10,27,22,34)( 4, 9,28,21,33)( 5,11,30,23,36)( 6,12,29,24,35)$
5A2 $5^{6},1^{6}$ $72$ $5$ $24$ $( 1,25,32, 8,20)( 2,26,31, 7,19)( 3,27,34,10,22)( 4,28,33, 9,21)( 5,30,36,11,23)( 6,29,35,12,24)$
6A $6^{6}$ $2$ $6$ $30$ $( 1, 5, 4, 2, 6, 3)( 7,12,10, 8,11, 9)(13,18,15,14,17,16)(19,24,22,20,23,21)(25,30,28,26,29,27)(31,35,34,32,36,33)$
6B $6^{6}$ $40$ $6$ $30$ $( 1,26,32, 2,25,31)( 3,28,34, 4,27,33)( 5,29,36, 6,30,35)( 7,20,14, 8,19,13)( 9,22,15,10,21,16)(11,24,18,12,23,17)$
6C $6^{3},2^{9}$ $40$ $6$ $24$ $( 1,31,25, 2,32,26)( 3,33,27, 4,34,28)( 5,35,30, 6,36,29)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
6D $6^{6}$ $80$ $6$ $30$ $( 1,30,21, 2,29,22)( 3,25,23, 4,26,24)( 5,28,19, 6,27,20)( 7,35,16, 8,36,15)( 9,31,17,10,32,18)(11,33,14,12,34,13)$
6E $6^{6}$ $80$ $6$ $30$ $( 1, 5, 4, 2, 6, 3)( 7,12,10, 8,11, 9)(13,18,15,14,17,16)(19,29,34,20,30,33)(21,26,35,22,25,36)(23,28,31,24,27,32)$
6F $6^{4},3^{4}$ $90$ $6$ $28$ $( 1,17, 4,13, 6,15)( 2,18, 3,14, 5,16)( 7,30,10,26,11,27)( 8,29, 9,25,12,28)(19,23,22)(20,24,21)(31,36,34)(32,35,33)$
6G $6^{6}$ $90$ $6$ $30$ $( 1, 5, 4, 2, 6, 3)( 7,24,10,20,11,21)( 8,23, 9,19,12,22)(13,18,15,14,17,16)(25,36,28,31,29,34)(26,35,27,32,30,33)$
6H $6^{6}$ $120$ $6$ $30$ $( 1,11,13, 5, 8,18)( 2,12,14, 6, 7,17)( 3, 9,16, 4,10,15)(19,29,31,24,26,35)(20,30,32,23,25,36)(21,27,33,22,28,34)$
6I $6^{4},3^{4}$ $120$ $6$ $28$ $( 1,25,13)( 2,26,14)( 3,30,16, 5,27,18)( 4,29,15, 6,28,17)( 7,19,31)( 8,20,32)( 9,24,33,12,21,35)(10,23,34,11,22,36)$
6J $6^{3},2^{9}$ $120$ $6$ $24$ $( 1, 2)( 3, 6)( 4, 5)( 7,13,31, 8,14,32)( 9,18,33,11,15,36)(10,17,34,12,16,35)(19,20)(21,23)(22,24)(25,26)(27,29)(28,30)$
6K $6^{2},3^{2},2^{6},1^{6}$ $120$ $6$ $20$ $( 1, 6)( 2, 5)( 7,11)( 8,12)(13,29,20,17,25,24)(14,30,19,18,26,23)(15,28,21)(16,27,22)(31,36)(32,35)$
10A1 $10^{3},2^{3}$ $72$ $10$ $30$ $( 1,26,32,19, 8, 2,25,31,20, 7)( 3,28,34,21,10, 4,27,33,22, 9)( 5,29,36,24,11, 6,30,35,23,12)(13,14)(15,16)(17,18)$
10A3 $10^{3},2^{3}$ $72$ $10$ $30$ $( 1,19,25, 7,32, 2,20,26, 8,31)( 3,21,27, 9,34, 4,22,28,10,33)( 5,24,30,12,36, 6,23,29,11,35)(13,14)(15,16)(17,18)$
10B1 $10^{3},2^{3}$ $216$ $10$ $30$ $( 1,19, 8,31,25, 2,20, 7,32,26)( 3,24,10,35,27, 6,22,12,34,29)( 4,23, 9,36,28, 5,21,11,33,30)(13,14)(15,18)(16,17)$
10B3 $10^{3},2^{3}$ $216$ $10$ $30$ $( 1,31,20,26, 8, 2,32,19,25, 7)( 3,35,22,29,10, 6,34,24,27,12)( 4,36,21,30, 9, 5,33,23,28,11)(13,14)(15,18)(16,17)$
10C1 $10^{2},5^{2},2^{2},1^{2}$ $216$ $10$ $28$ $( 1,29,13,35,20, 6,25,17,32,24)( 2,30,14,36,19, 5,26,18,31,23)( 3,27,16,34,22)( 4,28,15,33,21)( 7,11)( 8,12)$
10C3 $10^{2},5^{2},2^{2},1^{2}$ $216$ $10$ $28$ $( 1,35,25,24,13, 6,32,29,20,17)( 2,36,26,23,14, 5,31,30,19,18)( 3,34,27,22,16)( 4,33,28,21,15)( 7,11)( 8,12)$
12A $12^{2},6^{2}$ $180$ $12$ $32$ $( 1, 9,17,25, 4,12,13,28, 6, 8,15,29)( 2,10,18,26, 3,11,14,27, 5, 7,16,30)(19,34,23,31,22,36)(20,33,24,32,21,35)$
12B $12^{2},6^{2}$ $180$ $12$ $32$ $( 1,11,33,14, 6,10,32,18, 4, 7,35,16)( 2,12,34,13, 5, 9,31,17, 3, 8,36,15)(19,29,22,25,23,28)(20,30,21,26,24,27)$
15A1 $15^{2},3^{2}$ $144$ $15$ $32$ $( 1,12,21,32,29, 4, 8,24,33,25, 6, 9,20,35,28)( 2,11,22,31,30, 3, 7,23,34,26, 5,10,19,36,27)(13,17,15)(14,18,16)$
15A2 $15^{2},3^{2}$ $144$ $15$ $32$ $( 1,21,29, 8,33, 6,20,28,12,32, 4,24,25, 9,35)( 2,22,30, 7,34, 5,19,27,11,31, 3,23,26,10,36)(13,15,17)(14,16,18)$
30A1 $30,6$ $144$ $30$ $34$ $( 1,34,12,26,21, 5,32,10,29,19, 4,36, 8,27,24, 2,33,11,25,22, 6,31, 9,30,20, 3,35, 7,28,23)(13,16,17,14,15,18)$
30A7 $30,6$ $144$ $30$ $34$ $( 1,10,24,31,28, 5, 8,22,35,26, 4,11,20,34,29, 2, 9,23,32,27, 6, 7,21,36,25, 3,12,19,33,30)(13,16,17,14,15,18)$

Malle's constant $a(G)$:     $1/12$

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Character table

42 x 42 character table

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Regular extensions

Data not computed