Sylow subgroup of holomorph of cyclic group of odd prime-cube order
Let be an odd prime number. This group is defined as the -Sylow subgroup of the holomorph of the cyclic group of order . Equivalently, it is the semidirect product of the cyclic group of order and the cyclic group of order , where the generator of the latter acts on the former by the -power map.
It can be given by the explicit presentation:
Note that the case is different, because the automorphism structure of the cyclic group of order is different, with a non-cyclic 2-Sylow subgroup of automorphisms. See holomorph of Z8 for details on that group.
|order||semidirect product of groups of order and|
|exponent||element has order , no element of order|
|derived length||2||commutator subgroup cyclic of order|
|Frattini length||3||Frattini subgroup is , isomorphic to semidirect product of cyclic group of prime-square order and cyclic group of prime order|
|minimum size of generating set||2|
|rank as p-group||2||abelian subgroup|
|normal rank||2||abelian normal subgroup|
|characteristic rank||2||first omega subgroup is|
|group of prime power order||Yes|
|group of nilpotency class two||No|
|finite p-group that is not characteristic in any finite p-group properly containing it||Yes|
|regular p-group||Yes for|
|Lazard Lie group||Yes for|
Here, p</math> is the previously assigned value of the prime number, that we assume here to be odd. You can replace <tt>p by an actual numerical value of a prime or precede these commands by an assignment p := the value.
gap> C := CyclicGroup(p^3); <pc group of size 343 with 3 generators> gap> A := AutomorphismGroup(C); <group of size 294 with 6 generators> gap> S := SylowSubgroup(A,p); <group> gap> G := SemidirectProduct(S,C); <pc group with 5 generators>