Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#536 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| prime radical | C22×C3×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(3,4,5);(7,9)(10,11)(12,18)(13,15)(14,21)(16,20)(17,22)(19,23)(24,25);(10,12,15,20)(11,16,13,18)(14,19,22,25)(17,23,21,24);(6,7)(8,9)(10,13,15,11)(12,16,20,18)(14,21,22,17)(19,23,25,24);(1,2)(10,14)(11,17)(12,19)(13,21)(15,22)(16,23)(18,24)(20,25)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19907);