Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#563 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(3,4,5);(7,10)(8,12)(14,15,19,17)(16,22,24,20)(18,25,27,21)(23,28,26,29);(7,10)(11,13)(14,16,19,24)(15,20,17,22)(18,26,27,23)(21,29,25,28);(6,7)(8,11)(9,10)(12,13)(14,17,19,15)(16,20,24,22)(18,25,27,21)(23,28,26,29);(1,2)(6,8)(7,11)(9,12)(10,13)(14,18)(15,21)(16,23)(17,25)(19,27)(20,28)(22,29)(24,26)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19988);