Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#2916 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C2×C3 | C22×C4⋊C4 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×#41 ◁ G | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,6,10,7,11,5,20)(3,12,15,22,16,26,14,8)(4,9,19,24,17,23,18,30)(13,25,29,31,27,32,28,21)(33,34,35);(1,3,5,14,7,16,6,15)(2,8,10,12,11,22,20,26)(4,13,18,28,17,27,19,29)(9,21,24,25,23,31,30,32);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,25)(14,28)(15,29)(16,27)(20,30)(22,31)(26,32);(4,17)(9,23)(13,27)(18,19)(21,31)(24,30)(25,32)(28,29)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17356);