Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#3545 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,23)(21,24)(25,26,27);(3,8)(5,7)(9,11)(10,14)(12,21)(13,18)(15,17)(16,24)(19,22)(20,23);(1,3,4,8)(2,5,6,7)(10,15)(11,18)(16,20)(19,21);(2,6)(5,7)(9,12,13,22)(10,16,15,20)(11,19,18,21)(14,23,17,24);(10,15)(14,17)(16,20)(23,24)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20005);