Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×D4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | D4 |
| commutator | C4 | C22×C3×C8 |
| Frattini, cosocle | C42 | C23×C3 |
| socle | C23×C3 | C22⋊C4 |
| prime radical | C22×C3×D4 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C2×C3×C42 ◁ G | |
344 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(3,4,5);(7,10)(8,12)(11,13)(14,15,16,19,17,20,21,18);(6,7)(8,13)(9,11)(10,12)(14,15)(16,19)(17,20)(18,21)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4765);