Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C32 |
| socle | C2×C32 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C2×C32×C4 ◁ G | |
186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(3,10)(4,7)(5,9)(6,12)(8,14)(11,15)(13,16)(17,18,19);(1,3)(2,6)(4,11)(5,13)(7,10)(8,16)(9,15)(12,14)(20,21,22);(1,4,5,14)(2,7,9,8)(3,11,13,12)(6,10,15,16)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 832);