Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#2931 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C23 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C23×C4 |
| prime radical | C3×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(3,4)(5,6,8,11,9,12,16,19)(7,13,15,20,14,10,17,18)(21,22,23);(2,4)(7,14)(10,13)(15,17)(18,20);(5,6,8,11,9,12,16,19)(7,10,15,18,14,13,17,20);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,17)(19,20)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17371);