Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×D16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | D8 |
| commutator, Frattini, cosocle | C8 | C22×C32 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C32×C4 ◁ C32×C8 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3)(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22);(4,5,6)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 329);