Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#1416 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | D16 |
| commutator | C16 | C2×C3×C4 |
| Frattini, cosocle | C2×C16 | C22×C3 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×C16 ◁ G | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4)(6,11)(7,15)(8,17)(9,21)(12,27)(13,30)(14,26)(16,20)(18,34)(19,24)(22,28)(23,25)(29,35)(31,32)(33,36)(37,38,39);(5,6,10,14)(7,16,19,33)(8,20,22,36)(9,23,21,13)(11,24,26,15)(12,28,29,17)(18,35,31,27)(25,32,30,34)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 552);