The Frattini subgroup of a group $G$ is the intersection of the maximal subgroups of $G$, or $G$ itself in case $G$ has no maximal subgroups.

The Frattini subgroup is equal to the set of "non-generators" of $G$.  An element $g$ of $G$ is a non-generator if, whenever $G$ is generated by a set $S$ containing $g$, it is also generated by $S\setminus \left\{g\right\}$.

The Frattini subgroup of a finite group is nilpotent.

The FrattiniSubgroup( G ) command returns the Frattini subgroup of a group G.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{SmallGroup}\left(32\,5\right)\:$

$F≔\mathrm{FrattiniSubgroup}\left(G\right)$

$F≔\Phi \left(⟨\left(1\,2\,6\,11\,8\,12\,7\,3\right)\left(4\,15\,18\,30\,20\,31\,19\,16\right)\left(5\,10\,21\,27\,23\,28\,22\,14\right)\left(9\,24\,25\,32\,26\,29\,13\,17\right)\,\left(1\,4\right)\left(2\,9\right)\left(3\,13\right)\left(5\,17\right)\left(6\,18\right)\left(7\,19\right)\left(8\,20\right)\left(10\,15\right)\left(11\,25\right)\left(12\,26\right)\left(14\,16\right)\left(21\,24\right)\left(22\,29\right)\left(23\,32\right)\left(27\,30\right)\left(28\,31\right)\,\left(1\,5\right)\left(2\,10\right)\left(3\,14\right)\left(4\,17\right)\left(6\,21\right)\left(7\,22\right)\left(8\,23\right)\left(9\,15\right)\left(11\,27\right)\left(12\,28\right)\left(13\,16\right)\left(18\,24\right)\left(19\,29\right)\left(20\,32\right)\left(25\,30\right)\left(26\,31\right)\,\left(1\,6\,8\,7\right)\left(2\,11\,12\,3\right)\left(4\,18\,20\,19\right)\left(5\,21\,23\,22\right)\left(9\,25\,26\,13\right)\left(10\,27\,28\,14\right)\left(15\,30\,31\,16\right)\left(17\,24\,32\,29\right)\,\left(1\,8\right)\left(2\,12\right)\left(3\,11\right)\left(4\,20\right)\left(5\,23\right)\left(6\,7\right)\left(9\,26\right)\left(10\,28\right)\left(13\,25\right)\left(14\,27\right)\left(15\,31\right)\left(16\,30\right)\left(17\,32\right)\left(18\,19\right)\left(21\,22\right)\left(24\,29\right)⟩\right)$

$\mathrm{GroupOrder}\left(F\right)$

$8$

$\mathrm{IsNilpotent}\left(F\right)$

$\mathrm{true}$

$F≔\mathrm{FrattiniSubgroup}\left(\mathrm{DihedralGroup}\left(12\right)\right)$

$F≔\Phi \left({\mathbf{D}}_{12}\right)$

$\mathrm{GroupOrder}\left(F\right)$

$2$

$\mathrm{GroupOrder}\left(\mathrm{FrattiniSubgroup}\left(\mathrm{Alt}\left(4\right)\right)\right)$

$1$

$G≔\mathrm{QuasicyclicGroup}\left(7\right)$

$G≔{\mathbb{Z}}_{7^{\mathrm{\infty }}}$

$\mathrm{FrattiniSubgroup}\left(G\right)$

${\mathbb{Z}}_{7^{\mathrm{\infty }}}$

$F≔\mathrm{FrattiniSubgroup}\left(\mathrm{DirectProduct}\left(\mathrm{SemiDihedralGroup}\left(6\right)\,\mathrm{GL}\left(2\,3\right)\right)\right)$

$F≔\Phi \left(⟨\left(1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\,13\,14\,15\,16\,17\,18\,19\,20\,21\,22\,23\,24\right)\,\left(2\,12\right)\left(3\,23\right)\left(4\,10\right)\left(5\,21\right)\left(6\,8\right)\left(7\,19\right)\left(9\,17\right)\left(11\,15\right)\left(14\,24\right)\left(16\,22\right)\left(18\,20\right)\,\left(27\,30\right)\left(28\,31\right)\left(29\,32\right)\,\left(25\,30\,29\right)\left(26\,27\,31\right)⟩\right)$

$\mathrm{AreIsomorphic}\left(F\,\mathrm{DirectProduct}\left(\mathrm{FrattiniSubgroup}\left(\mathrm{SemiDihedralGroup}\left(6\right)\right)\,\mathrm{FrattiniSubgroup}\left(\mathrm{GL}\left(2\,3\right)\right)\right)\right)$

$\mathrm{true}$

The GroupTheory[FrattiniSubgroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.