Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#1419 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #357 |
| commutator | C16 | C2×C3×C4 |
| Frattini, cosocle | C2×C16 | C22×C3 |
| prime radical | C2×C3×D16 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×C16 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,4,10,7,12,6,23)(3,13,16,30,19,8,18,24)(5,20,15,27,22,28,21,11)(9,26,25,14,29,32,31,17)(33,34,35);(2,8)(3,14)(4,6)(5,21)(9,27)(10,30)(11,29)(12,13)(15,22)(16,26)(17,19)(18,32)(20,25)(23,24)(28,31)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 555);