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×#3551 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
1050 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)(11,19)(12,18)(13,17)(14,16)(15,20)(21,23)(22,24)(25,26,27);(3,8)(5,7)(9,11)(10,15)(12,22)(13,21)(14,20)(16,19)(17,24)(18,23);(1,3,4,8)(2,5,6,7)(10,16)(11,20)(17,18)(21,22);(2,6)(5,7)(9,12,14,13)(10,17,16,18)(11,21,20,22)(15,23,19,24);(9,13,14,12)(10,18,16,17)(11,21,20,22)(15,23,19,24)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20011);