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Properties

Label 36T2844
Degree $36$
Order $1944$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times C_3^3:A_4$

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Copy content magma:G := TransitiveGroup(36, 2844);
 

Group invariants

Abstract group:  $S_3\times C_3^3:A_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $1944=2^{3} \cdot 3^{5}$
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 $t$:  $2844$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,20,2,19,3,21)(4,22,6,24,5,23)(7,27,8,26,9,25)(10,28,11,30,12,29)(13,32)(14,33)(15,31)(16,36)(17,35)(18,34)$, $(1,17,13,21,25,24)(2,18,15,19,26,23)(3,16,14,20,27,22)(4,30,8,32,10,35)(5,28,7,33,11,34)(6,29,9,31,12,36)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$12$:  $A_4$
$18$:  $S_3\times C_3$
$24$:  $A_4\times C_2$
$72$:  12T43
$324$:  $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$
$648$:  18T199

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$, $A_4\times C_2$

Degree 9: None

Degree 12: $A_4\times C_2$

Degree 18: None

Low degree siblings

18T347, 24T4982 x 2, 24T4985, 27T399, 36T2770 x 2, 36T2771, 36T2973, 36T2974 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 $2^{18}$ $3$ $2$ $18$ $( 1,19)( 2,20)( 3,21)( 4,33)( 5,31)( 6,32)( 7,36)( 8,34)( 9,35)(10,28)(11,29)(12,30)(13,23)(14,24)(15,22)(16,26)(17,27)(18,25)$
2B $2^{18}$ $27$ $2$ $18$ $( 1,12)( 2,10)( 3,11)( 4,32)( 5,31)( 6,33)( 7,17)( 8,18)( 9,16)(13,22)(14,24)(15,23)(19,30)(20,28)(21,29)(25,34)(26,35)(27,36)$
2C $2^{16},1^{4}$ $81$ $2$ $16$ $( 1,28)( 2,29)( 3,30)( 4, 5)( 7,27)( 8,25)( 9,26)(10,20)(11,21)(12,19)(13,14)(16,34)(17,35)(18,36)(22,24)(32,33)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 2, 3)( 4, 5, 6)( 7, 9, 8)(10,11,12)(13,15,14)(16,18,17)(19,21,20)(22,23,24)(25,26,27)(28,30,29)(31,33,32)(34,35,36)$
3B1 $3^{6},1^{18}$ $4$ $3$ $12$ $( 1, 2, 3)( 7, 9, 8)(13,14,15)(19,20,21)(22,23,24)(34,36,35)$
3B-1 $3^{6},1^{18}$ $4$ $3$ $12$ $( 1, 3, 2)( 7, 8, 9)(13,15,14)(19,21,20)(22,24,23)(34,35,36)$
3C $3^{8},1^{12}$ $6$ $3$ $16$ $( 1, 2, 3)( 4, 6, 5)(10,12,11)(13,14,15)(19,20,21)(22,23,24)(28,30,29)(31,33,32)$
3D1 $3^{9},1^{9}$ $8$ $3$ $18$ $( 1, 2, 3)(13,15,14)(16,18,17)(19,21,20)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,36,35)$
3D-1 $3^{9},1^{9}$ $8$ $3$ $18$ $( 4, 5, 6)(10,11,12)(13,14,15)(16,18,17)(19,20,21)(25,26,27)(28,30,29)(31,33,32)(34,36,35)$
3E $3^{10},1^{6}$ $12$ $3$ $20$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(16,18,17)(19,20,21)(25,27,26)(28,30,29)(31,32,33)(34,35,36)$
3F $3^{8},1^{12}$ $12$ $3$ $16$ $( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,14,15)(16,18,17)(22,24,23)(31,32,33)$
3G $3^{7},1^{15}$ $24$ $3$ $14$ $( 7, 8, 9)(10,12,11)(13,15,14)(19,20,21)(22,23,24)(25,27,26)(31,32,33)$
3H1 $3^{12}$ $36$ $3$ $24$ $( 1, 9,24)( 2, 8,22)( 3, 7,23)( 4,28,17)( 5,30,16)( 6,29,18)(10,27,33)(11,25,32)(12,26,31)(13,21,36)(14,19,35)(15,20,34)$
3H-1 $3^{12}$ $36$ $3$ $24$ $( 1,24, 9)( 2,22, 8)( 3,23, 7)( 4,17,28)( 5,16,30)( 6,18,29)(10,33,27)(11,32,25)(12,31,26)(13,36,21)(14,35,19)(15,34,20)$
3I1 $3^{12}$ $72$ $3$ $24$ $( 1, 4,36)( 2, 5,34)( 3, 6,35)( 7,21,31)( 8,19,32)( 9,20,33)(10,15,16)(11,14,18)(12,13,17)(22,27,29)(23,25,28)(24,26,30)$
3I-1 $3^{12}$ $72$ $3$ $24$ $( 1,36, 4)( 2,34, 5)( 3,35, 6)( 7,31,21)( 8,32,19)( 9,33,20)(10,16,15)(11,18,14)(12,17,13)(22,29,27)(23,28,25)(24,30,26)$
6A1 $6^{3},2^{9}$ $12$ $6$ $24$ $( 1,19, 2,20, 3,21)( 4,31)( 5,32)( 6,33)( 7,36, 9,35, 8,34)(10,29)(11,30)(12,28)(13,24,14,22,15,23)(16,25)(17,26)(18,27)$
6A-1 $6^{3},2^{9}$ $12$ $6$ $24$ $( 1,21, 3,20, 2,19)( 4,31)( 5,32)( 6,33)( 7,34, 8,35, 9,36)(10,29)(11,30)(12,28)(13,23,15,22,14,24)(16,25)(17,26)(18,27)$
6B $6^{4},2^{6}$ $18$ $6$ $26$ $( 1,21, 2,19, 3,20)( 4,31, 6,33, 5,32)( 7,36)( 8,34)( 9,35)(10,29,12,28,11,30)(13,22,14,23,15,24)(16,26)(17,27)(18,25)$
6C $6^{5},2^{3}$ $36$ $6$ $28$ $( 1,21, 2,19, 3,20)( 4,32, 5,33, 6,31)( 7,35, 8,36, 9,34)(10,29,12,28,11,30)(13,23)(14,24)(15,22)(16,27,18,26,17,25)$
6D $6^{6}$ $54$ $6$ $30$ $( 1,11, 2,12, 3,10)( 4,33, 5,32, 6,31)( 7,18, 9,17, 8,16)(13,24,15,22,14,23)(19,28,21,30,20,29)(25,36,26,34,27,35)$
6E $6^{4},2^{6}$ $54$ $6$ $26$ $( 1,12, 2,10, 3,11)( 4,31)( 5,33)( 6,32)( 7,16, 9,18, 8,17)(13,22)(14,24)(15,23)(19,30,20,28,21,29)(25,34,27,36,26,35)$
6F $6^{4},2^{6}$ $108$ $6$ $26$ $( 1,10, 2,11, 3,12)( 4,33, 6,31, 5,32)( 7,18, 9,17, 8,16)(13,24,14,23,15,22)(19,29)(20,30)(21,28)(25,34)(26,35)(27,36)$
6G1 $6^{6}$ $108$ $6$ $30$ $( 1,15, 9,20,24,34)( 2,13, 8,21,22,36)( 3,14, 7,19,23,35)( 4,26,28,31,17,12)( 5,25,30,32,16,11)( 6,27,29,33,18,10)$
6G-1 $6^{6}$ $108$ $6$ $30$ $( 1,34,24,20, 9,15)( 2,36,22,21, 8,13)( 3,35,23,19, 7,14)( 4,12,17,31,28,26)( 5,11,16,32,30,25)( 6,10,18,33,29,27)$
6H $6^{4},2^{4},1^{4}$ $162$ $6$ $24$ $( 1,29, 3,28, 2,30)( 4, 5)( 7,26, 8,27, 9,25)(10,21,12,20,11,19)(13,14)(16,36,17,34,18,35)(22,24)(32,33)$
9A1 $9^{2},3^{6}$ $36$ $9$ $28$ $( 1, 8,23, 2, 7,24, 3, 9,22)( 4,28,16)( 5,30,18)( 6,29,17)(10,26,33)(11,27,32)(12,25,31)(13,20,36,14,21,35,15,19,34)$
9A-1 $9^{2},3^{6}$ $36$ $9$ $28$ $( 1,22, 9, 3,24, 7, 2,23, 8)( 4,16,28)( 5,18,30)( 6,17,29)(10,33,26)(11,32,27)(12,31,25)(13,34,19,15,35,21,14,36,20)$
9B1 $9^{2},3^{6}$ $36$ $9$ $28$ $( 1, 5,36)( 2, 6,34)( 3, 4,35)( 7,19,31)( 8,20,32)( 9,21,33)(10,15,17,11,14,16,12,13,18)(22,27,29,24,26,30,23,25,28)$
9B-1 $9^{2},3^{6}$ $36$ $9$ $28$ $( 1,36, 5)( 2,34, 6)( 3,35, 4)( 7,31,19)( 8,32,20)( 9,33,21)(10,18,13,12,16,14,11,17,15)(22,28,25,23,30,26,24,29,27)$
9C1 $9^{2},3^{6}$ $72$ $9$ $28$ $( 1,23, 9, 2,24, 8, 3,22, 7)( 4,16,30)( 5,18,29)( 6,17,28)(10,31,26)(11,33,27)(12,32,25)(13,36,21,14,35,19,15,34,20)$
9C-1 $9^{2},3^{6}$ $72$ $9$ $28$ $( 1, 7,22, 3, 8,24, 2, 9,23)( 4,30,16)( 5,29,18)( 6,28,17)(10,26,31)(11,27,33)(12,25,32)(13,20,34,15,19,35,14,21,36)$
9D1 $9^{2},3^{6}$ $72$ $9$ $28$ $( 1, 6,36)( 2, 4,34)( 3, 5,35)( 7,21,33)( 8,19,31)( 9,20,32)(10,14,17,12,15,18,11,13,16)(22,26,30,23,27,29,24,25,28)$
9D-1 $9^{2},3^{6}$ $72$ $9$ $28$ $( 1,36, 6)( 2,34, 4)( 3,35, 5)( 7,33,21)( 8,31,19)( 9,32,20)(10,16,13,11,18,15,12,17,14)(22,28,25,24,29,27,23,30,26)$
18A1 $18,6^{3}$ $108$ $18$ $32$ $( 1,15, 8,19,23,34, 2,13, 7,20,24,36, 3,14, 9,21,22,35)( 4,25,28,31,16,12)( 5,27,30,32,18,11)( 6,26,29,33,17,10)$
18A-1 $18,6^{3}$ $108$ $18$ $32$ $( 1,35,22,21, 9,14, 3,36,24,20, 7,13, 2,34,23,19, 8,15)( 4,12,16,31,28,25)( 5,11,18,32,30,27)( 6,10,17,33,29,26)$
18B1 $18,6^{3}$ $108$ $18$ $32$ $( 1, 9, 5,21,36,33)( 2, 7, 6,19,34,31)( 3, 8, 4,20,35,32)(10,27,15,29,17,24,11,26,14,30,16,23,12,25,13,28,18,22)$
18B-1 $18,6^{3}$ $108$ $18$ $32$ $( 1,33,36,21, 5, 9)( 2,31,34,19, 6, 7)( 3,32,35,20, 4, 8)(10,22,18,28,13,25,12,23,16,30,14,26,11,24,17,29,15,27)$

Malle's constant $a(G)$:     $1/12$

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Character table

39 x 39 character table

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Regular extensions

Data not computed