Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#3111 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C3×#449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,15)(13,16)(14,20)(17,21)(19,23)(22,24)(25,26,27);(2,5)(6,8)(9,11,13,20)(10,14,16,18)(12,19,21,24)(15,22,17,23);(10,15)(12,21)(14,22)(16,17)(18,23)(19,24);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,15,16,17)(11,19,20,24)(14,22,18,23)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17551);