Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#464 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(3,4,5);(7,9)(10,11)(12,18)(13,15)(14,22)(16,20)(17,21)(19,25)(23,24);(1,2)(6,7,8,9)(11,15)(16,24)(17,22)(18,25);(1,2)(10,12,13,19)(11,16,15,24)(14,23,21,20)(17,25,22,18)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17474);