Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C32×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | D4 |
| commutator | C4 | C2×C32×C4 |
| Frattini, cosocle | C2×C4 | C22×C32 |
| socle | C2×C32 | C4⋊C4 |
| prime radical | C22×C32 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C2×C32×C4 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7)(17,18,19);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16)(20,21,22)126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 326);