This is a method to construct up to isomorphism the groups of order
`p ^{n} cdotq` for different primes

Note that all functions described in this chapter rely on an
efficient method for `AutomorphismGroup`

for `p`-groups. Such
a method is provided in the forthcoming share package AutPGrp.
Thus it is useful to install and load this share package before
using the functions described in this chapter.

`CyclicSplitExtensionMethod( `

`, `

`, `

` ) F`

`CyclicSplitExtensionMethod( `

`, `

`, `

`, `

` ) F`

Clearly, each of the computed groups is a split extension of a group
of order `p ^{n}` and the cyclic group of order

As in Chapter The Frattini Extension Method all groups are
described as codes. Setting `uncoded` to true, the function
will return pc groups instead.

If one wants to construct the groups of order `p ^{n} cdotq`
for fixed

gap> CyclicSplitExtensionMethod( 2,2,7, true ); rec( up := [ ], down := [ <pc group of size 28 with 3 generators>, <pc group of size 28 with 3 generators> ], both := [ <pc group of size 28 with 3 generators>, <pc group of size 28 with 3 generators> ] ) gap> CyclicSplitExtensionMethod( 2,2,[3,5], true ); rec( up := [ <pc group of size 12 with 3 generators> ], down := [ <pc group of size 12 with 3 generators>, <pc group of size 20 with 3 generators>, <pc group of size 20 with 3 generators>, <pc group of size 12 with 3 generators>, <pc group of size 20 with 3 generators> ], both := [ <pc group of size 12 with 3 generators>, <pc group of size 20 with 3 generators>, <pc group of size 12 with 3 generators>, <pc group of size 20 with 3 generators> ] )

Note that the function `CyclicSplitExtensionMethod`

requires that
the groups of order `p ^{n}` are given within the SmallGroups Library.

It is possible to construct the cyclic extensions of a single group
of order `p ^{n}` only. The output is as above.

`CyclicSplitExtensions( `

`, `

` ) F`

`CyclicSplitExtensions( `

`, `

`, `

` ) F`

Moreover, the computation of the record entry `up` and the record
entry `down` can be separated by using the following functions.

`CyclicSplitExtensionsUp( `

`, `

` ) F`

`CyclicSplitExtensionsUp( `

`, `

`, `

` ) F`

`CyclicSplitExtensionsDown( `

`, `

` ) F`

`CyclicSplitExtensionsDown( `

`, `

`, `

` ) F`

The input for these functions is the same as above. The first
function returns a list of groups with one normal subgroup of order
`p ^{n}` and the second a list of groups with one normal subgroup of order

gap> G := SmallGroup( 16, 10 );; gap> CyclicSplitExtensionsUp( G, 3, true ); [ <pc group with 5 generators> ] gap> G := SylowSubgroup( SymmetricGroup(4), 2); Group([ (1,2), (3,4), (1,3)(2,4) ]) gap> CyclicSplitExtensionsDown( G, 3 ); [ rec( code := 6562689, order := 24 ), rec( code := 2837724033, order := 24 ) ]

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grpconst manual

Mai 2012