Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×#1321 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #99 |
| commutator | C2×C4 | C2×C3×C8 |
| Frattini, cosocle | C2×M16 | C22×C3 |
| socle | C2×C3 | #323 |
| prime radical | C2×C3×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ G | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,5,10,6,9,20,26,7,11,21,28,4,16,18,27)(3,12,14,29,13,8,22,23,15,25,19,31,17,24,30,32)(33,34,35);(1,3)(2,8)(4,17)(5,19)(6,13)(7,15)(9,25)(10,23)(11,24)(12,16)(14,21)(18,22)(20,30)(26,31)(27,29)(28,32)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 457);