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Properties

Label 36T43671
Degree $36$
Order $2239488$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_6^4.D_6^2:D_6$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 43671);
 

Group invariants

Abstract group:  $C_6^4.D_6^2:D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $2239488=2^{10} \cdot 3^{7}$
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 $t$:  $43671$
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:  $(1,13,6,17,4,16)(2,14,5,18,3,15)(7,10)(8,9)(19,25,21,28,24,30,20,26,22,27,23,29)(31,35)(32,36)$, $(1,18,5,16,4,14,2,17,6,15,3,13)(7,8)(9,11)(10,12)(19,26,22,28,24,29)(20,25,21,27,23,30)(31,34)(32,33)$, $(1,30,5,26)(2,29,6,25)(3,28,4,27)(7,23,9,19)(8,24,10,20)(11,21,12,22)(13,33,18,31)(14,34,17,32)(15,35,16,36)$, $(1,17,4,13,6,16)(2,18,3,14,5,15)(7,8)(9,11)(10,12)(19,29,22,28,24,26)(20,30,21,27,23,25)(31,36,34,32,35,33)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$6$:  $S_3$
$8$:  $D_{4}$ x 12, $C_2^3$ x 15
$12$:  $D_{6}$ x 7
$16$:  $D_4\times C_2$ x 18, $C_2^4$
$24$:  $S_4$, $S_3 \times C_2^2$ x 7
$32$:  $C_2^2 \wr C_2$ x 4, $C_2^2 \times D_4$ x 3
$48$:  $S_4\times C_2$ x 7, 12T28 x 6, 24T30
$64$:  16T105
$72$:  $C_3^2:D_4$
$96$:  12T48 x 7, 24T143 x 3
$144$:  12T77 x 3
$192$:  $V_4^2:(S_3\times C_2)$, 12T86 x 6, 24T360, 24T400
$288$:  12T125 x 2, 24T654
$384$:  12T136 x 3, 24T1076 x 3
$432$:  12T156
$576$:  24T1475
$768$:  12T186 x 6, 24T2202, 24T2481
$864$:  24T2646
$1536$:  24T4787 x 3
$1728$:  24T4923, 24T4943
$3072$:  24T7067
$3456$:  36T4476
$3888$:  18T440
$6912$:  24T9626, 36T6999
$7776$:  36T7260
$13824$:  36T9778
$15552$:  36T10081, 36T10082
$27648$:  36T13216
$31104$:  36T13539
$34992$:  18T675
$62208$:  36T17293, 36T17301
$69984$:  36T17550
$124416$:  36T20815
$139968$:  36T21097, 36T21098
$248832$:  36T25133
$279936$:  36T25385
$559872$:  36T30523, 36T30531
$1119744$:  36T36567

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$

Degree 9: None

Degree 12: 12T193

Degree 18: 18T675

Low degree siblings

36T43671 x 95

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Conjugacy classes not computed

Copy content magma:ConjugacyClasses(G);
 

Character table

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed