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Properties

Label 36T20490
Degree $36$
Order $110592$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4^2.C_2\wr D_6$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 20490);
 

Group invariants

Abstract group:  $A_4^2.C_2\wr D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $110592=2^{12} \cdot 3^{3}$
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 $t$:  $20490$
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:  $(1,12)(2,11)(3,9)(4,10)(5,8)(6,7)(13,24,14,23)(15,21,16,22)(17,20,18,19)(25,35,26,36)(27,34,28,33)(29,32,30,31)$, $(1,24,16,31,29,9,2,23,15,32,30,10)(3,20,17,34,25,11,4,19,18,33,26,12)(5,21,13,36,27,8,6,22,14,35,28,7)$, $(1,10,26,33,14,21)(2,9,25,34,13,22)(3,8,28,31,16,19,4,7,27,32,15,20)(5,12,29,36,18,23)(6,11,30,35,17,24)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$3$:  $C_3$
$4$:  $C_2^2$ x 7
$6$:  $S_3$ x 2, $C_6$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$12$:  $D_{6}$ x 6, $C_6\times C_2$ x 7
$16$:  $D_4\times C_2$
$18$:  $S_3\times C_3$ x 2
$24$:  $S_4$, $S_3 \times C_2^2$ x 2, $(C_6\times C_2):C_2$ x 2, $D_4 \times C_3$ x 2, 24T3
$36$:  $S_3^2$, $C_6\times S_3$ x 6
$48$:  $S_4\times C_2$ x 3, 12T28, 24T25, 24T38
$72$:  12T37, 12T42 x 2, 12T45, 24T68 x 2
$96$:  12T48
$108$:  12T70
$144$:  12T83, 18T61 x 3, 24T204, 24T208, 24T248
$192$:  $V_4^2:(S_3\times C_2)$, 12T86
$216$:  24T547
$288$:  $A_4\wr C_2$, 18T111, 36T330
$384$:  12T136
$432$:  24T1280, 24T1328
$576$:  12T158 x 3, 24T1496, 24T1497, 36T739, 36T771
$768$:  12T186
$864$:  36T1298
$1152$:  12T208 x 2, 24T2772, 36T1629, 36T1654
$1728$:  24T4931, 24T4944, 36T2390
$2304$:  24T5101, 36T3070, 36T3112
$3456$:  36T4310, 36T4315
$6912$:  32T399497, 36T6859, 36T6860
$13824$:  36T9644
$27648$:  32T1123267, 36T13105
$55296$:  36T16925

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $D_{6}$, $S_3\times C_3$

Degree 9: None

Degree 12: None

Degree 18: 18T46

Low degree siblings

36T20490 x 7

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