Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#452 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
760 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);(1,2)(7,9)(10,11,15,13)(12,18,21,25)(14,22,24,17)(16,23,19,20);(10,12)(11,16)(13,19)(14,23)(15,21)(17,25)(18,22)(20,24);(1,2)(6,7)(8,9)(10,13,15,11)(12,19,21,16)(14,22,24,17)(18,20,25,23)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17441);