# SmallGroup(32,33)

View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

This group is a semidirect product $(Z_4 \times Z_4) \rtimes Z_2$, with the following presentation:

$\langle a,x,y \mid a^4 = b^4 = x^2 = e, ab = ba, xax^{-1} = ab^2, xbx^{-1} = a^2b^3 \rangle$

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#Arithmetic functions

## Subgroup-defining functions

Subgroup-defining function Isomorphism class Corresponding quotient-defining function Isomorphism class of quotient
center Klein four-group inner automorphism group elementary abelian group:E8
derived subgroup Klein four-group abelianization elementary abelian group:E8
Frattini subgroup Klein four-group Frattini quotient elementary abelian group:E8
socle Klein four-group  ? elementary abelian group:E8
join of abelian subgroups of maximum order direct product of Z4 and Z4  ? cyclic group:Z2
join of abelian subgroups of maximum rank elementary abelian group:E8  ? Klein four-group
ZJ-subgroup direct product of Z4 and Z4  ? cyclic group:Z2

## GAP implementation

### Group ID

This finite group has order 32 and has ID 33 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(32,33)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(32,33);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [32,33]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.