Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#2262 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | D16 |
| commutator, Frattini, cosocle | C16 | C23×C3 |
| socle | C2×C3 | C2×D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×C16 ◁ G | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 67 points:
(1,2)(3,14)(4,10)(5,21)(6,27)(7,30)(8,13)(9,16)(11,24)(12,23)(15,41)(17,46)(18,43)(19,48)(20,40)(22,39)(25,57)(26,52)(28,37)(29,59)(31,42)(32,44)(33,49)(34,53)(35,45)(36,50)(38,56)(47,63)(51,55)(54,61)(58,62)(60,64)(65,66,67);(1,3)(2,9)(4,15)(5,22)(6,28)(7,31)(8,18)(10,32)(11,37)(12,39)(13,35)(14,21)(16,24)(17,23)(19,49)(20,47)(25,58)(26,54)(27,33)(29,60)(30,34)(36,63)(38,64)(40,61)(41,48)(42,57)(43,52)(44,50)(45,56)(46,51)(53,55)(59,62);(1,4,8,7)(2,10,13,30)(3,15,18,31)(5,19,26,25)(6,20,29,23)(9,32,35,34)(11,36,38,51)(12,27,40,59)(14,41,43,42)(16,44,45,53)(17,28,47,60)(21,48,52,57)(22,49,54,58)(24,50,56,55)(33,61,62,39)(37,63,64,46)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5553);