Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#3518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,22,15)(12,17,13,16)(20,23,21,24)(25,26,27);(9,11,14,22)(10,15,18,19)(12,20,13,21)(16,23,17,24);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24);(9,12,14,13)(10,17,18,16)(11,21,22,20)(15,23,19,24);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,17,18,16)(11,21,22,20)(15,24,19,23)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19978);