I need to construct the following group in Magma: given $H=(C_2)^3 \rtimes (C_7 \rtimes C_3) \times C_3$ (so $\operatorname{SmallGroup}(168,43)\times C_3$), there is a non-split central extension by $C_3$. We call it $\hat{H} = (C_2)^3 \rtimes (C_7 \rtimes 3^{1+2}_+)$.
Given $\hat{N}=A_4=\operatorname{SmallGroup}(12,3)$ we can consider its normal subgroup $N = C_2 \times C_2$.
I want to build an extension of $H \times N$ by $C_3$ that contains both $\hat{H}$ and $\hat{N}$ (so I am identifying the two $C_3$s that give these actions)
When the extensions are split, I usually do this kind of thing by constructing the direct product $G= \hat{H} \times \hat{N}$ and then taking normal subgroups $H$ and $N$ and elements $x \in \hat{H}\setminus H$ and $y \in \hat{N}\setminus N$ that give the action of the corresponding extensions, and using $$\operatorname{sub}<G | H, N, x*y>;$$
but in this case I honestly have no clue since $C_3$ does not "act" on $H$.
An answer in GAP would also be fine, as I just need to examine certain invariants of this group.
Best Answer
I just discovered that Magma supports "GroupNames", so I tried the following instruction
G:=Group("C2^5:C7:He3");and then I checked some properties:
Order(G); Center(G); Centraliser(G,Sylow(G,7)); Centraliser(G,Sylow(G,2)); T:=Sylow(G,3); Centraliser(G,T.1); Centraliser(G,T.2); Centraliser(G,T.1*T.2);...and it seems that this is indeed my group.