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Properties

Label 36T69301
Degree $36$
Order $26873856$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^8:C_2^3.D_4^2:D_4$

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Copy content magma:G := TransitiveGroup(36, 69301);
 

Group invariants

Abstract group:  $C_3^8:C_2^3.D_4^2:D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $26873856=2^{12} \cdot 3^{8}$
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 $t$:  $69301$
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:  $(1,7,9,3)(2,4,8,6)(10,31,14,28,17,36,13,30)(11,32,12,33,16,35,18,34)(15,29)(19,24,26)(20,22,27)(21,23,25)$, $(1,29)(2,33,3,34)(4,32,7,35)(5,36,9,31)(6,28,8,30)(10,23,13,26,17,27,14,24)(11,21,18,22,16,20,12,25)(15,19)$, $(1,14,23,36,9,17,26,31,2,13,25,32,6,10,22,34)(3,15,21,28)(4,12,27,33,7,16,19,30,8,18,24,35,5,11,20,29)$, $(1,31,25,15,9,36,24,10)(2,34,19,11,8,33,21,14)(3,28,22,16,7,30,27,18)(4,35,26,12,6,32,23,13)(5,29,20,17)$
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
$8$:  $D_{4}$ x 28, $C_2^3$ x 15
$16$:  $QD_{16}$ x 4, $D_4\times C_2$ x 42, $C_2^4$
$32$:  $Z_8 : Z_8^\times$ x 6, $C_2^2 \wr C_2$ x 28, $C_2^2 \times D_4$ x 7, 16T48 x 6
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T89 x 3, 16T105 x 7, 16T155 x 12, 32T277
$128$:  $C_2 \wr C_2\wr C_2$ x 4, 16T241, 16T245, 16T255 x 3, 16T325, 32T1254 x 3
$256$:  16T509 x 2, 32T4223, 32T4446, 32T5588
$512$:  16T827, 16T919, 32T18625, 32T20853
$1024$:  32T40644, 32T66057
$4096$:  32T233153

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: None

Degree 9: None

Degree 12: None

Degree 18: None

Low degree siblings

There are no siblings 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