Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#3371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C3×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×Q8 ◁ G | |
344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(2,5)(6,7)(9,10)(11,18)(12,15)(13,17)(14,21)(16,22)(19,24)(20,23)(25,26,27);(1,2,4,5)(3,7,8,6)(11,19)(14,23)(18,24)(20,21);(9,11,13,19)(10,14,17,23)(12,20,22,21)(15,18,16,24);(1,3,4,8)(2,6,5,7)(9,12,13,22)(10,15,17,16)(11,20,19,21)(14,18,23,24)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17811);