Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#400 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(3,4,5);(12,16)(14,17)(18,19,21,22)(20,24,25,23);(1,2)(7,9)(10,11,13,15)(12,14,16,17)(19,22)(23,24);(1,2)(6,7)(8,9)(10,12)(11,14)(13,16)(15,17)(18,20,21,25)(19,23,22,24)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16834);