Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#2263 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×D16 |
| commutator, Frattini, cosocle | C16 | C23×C3 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×C16 ◁ G | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(3,7)(4,16)(5,19)(6,8)(9,12)(10,13)(11,17)(14,26)(15,20)(18,32)(21,28)(22,27)(23,29)(24,30)(25,31)(33,34,35);(2,7)(3,12)(4,13)(5,17)(8,20)(9,10)(11,26)(14,31)(15,29)(16,21)(18,30)(19,22)(23,24)(25,32)(27,28);(2,8)(7,20)(9,23)(10,24)(11,19)(16,32)(21,25)(22,26)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5554);