The Lie group $G_2$ is one (or rather: three) of the exceptional Lie groups. One way to characterize it is as the automorphism group of the octonions. Another way to characterize it is as the subgroup of the general linear group $GL(7)$ of those elements that preserve the canonical differential 3-form $\langle ,(-)\times (-) \rangle$ on the Cartesian space $\mathbb{R}^7$. As such, the group $G_2$ is a higher analog of the symplectic group (which is the group that preserves a canonical 2-form on any $\mathbb{R}^{2n}$), obtained by passing from symplectic geometry to 2-plectic geometry.
On the Cartesian space $\mathbb{R}^7$ consider the associative 3-form, the constant differential 3-form $\omega \in \Omega^3(\mathbb{R}^7)$ given on tangent vectors $u,v,w \in \mathbb{R}^7$ by
where
$\langle -,-\rangle$ is the canonical bilinear form
$(-)\times(-)$ is the cross product of vectors.
Then the group $G_2 \hookrightarrow GL(7)$ is the subgroup of the general linear group acting on $\mathbb{R}^7$ which preserves the canonical orientation and preserves this 3-form $\omega$. Equivalently, it is the subgroup preserving the orientation and the Hodge dual differential 4-form $\star \omega$.
See for instance the introduction of (Joyce).
The inclusion $G_2 \hookrightarrow GL(7)$ of def. 1 factors through the special orthogonal group
The 3-form $\omega$ from def. 1 we may regard as equipping $\mathbb{R}^7$ with 2-plectic structure. From this point of view $G_2$ is the linear subgroup of the 2-plectomorphism group, hence (up to the translations) the image of the Heisenberg group of $(\mathbb{R}^7, \omega)$ in the symplectomorphism group.
Or, dually, we may regard the 4-form $\star \omega$ of def. 1 as being a 3-plectic structure and $G_2$ correspondingly as the linear part in the 3-plectomorphism group of $\mathbb{R}^7$.
G2, F4,
Surveys are in
The definitions are reviewed for instance in
Discussion in terms of the Heisenberg group in 2-plectic geometry is in
Cohomological properties are discussed in
Discussion of Yang-Mills theory with $G_2$ as gauge group is in