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×#3567 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2)(3,9)(4,12)(5,8)(6,11)(7,14)(10,15)(13,16)(17,18)(19,23)(20,29)(21,25)(22,24)(26,31)(27,32)(28,30)(33,34,35);(1,3)(2,6)(4,13)(5,11)(7,16)(8,9)(10,14)(12,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32);(1,4,5,14)(2,7,8,12)(3,10,11,13)(6,15,9,16)(18,24)(19,26)(25,29)(27,28);(2,8)(3,11)(7,12)(10,13)(17,20,22,21)(18,25,24,29)(19,27,26,28)(23,30,31,32);(2,8)(6,9)(7,12)(15,16)(17,21,22,20)(18,25,24,29)(19,28,26,27)(23,30,31,32)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20027);