Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| direct product | C3×SD128 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | D32 |
| commutator, Frattini, cosocle | C32 | C22×C3 |
| prime radical | C3×D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×C16 ◁ C3×C32 ◁ G | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 67 points:
(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67);(5,35)(6,66)(7,33)(8,64)(9,31)(10,62)(11,29)(12,60)(13,27)(14,58)(15,25)(16,56)(17,23)(18,54)(19,21)(20,52)(22,50)(24,48)(26,46)(28,44)(30,42)(32,40)(34,38)(37,67)(39,65)(41,63)(43,61)(45,59)(47,57)(49,55)(51,53)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 566);