Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#2209 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8 ◁ G | |
154 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 39 points:
(2,4)(5,6,10,14)(7,15,18,34)(8,19,23,33)(9,24,21,13)(11,26,28,16)(12,29,31,22)(17,35,25,30)(20,32,36,27)(37,38,39);(1,2,3,4);(5,7,10,18)(6,11,14,28)(8,20,23,36)(9,25,21,17)(12,30,31,35)(13,32,24,27)(15,33,34,19)(16,29,26,22)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5500);