Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | Dic3×#125 |
| Sylow | C3 ⋊ C4×#125 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C2×C3×#125 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | C2×#125 | D3 |
| hypocenter | C3 | C4×#125 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
484 conjugacy classes of subgroups; only the 249 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 249 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(9,13)(12,15)(16,18,19,22)(17,21,20,23);(8,10,11,14)(9,12,13,15)(17,20)(21,23);(8,9)(10,12)(11,13)(14,15)(16,17,19,20)(18,21,22,23);(2,3)(4,5,6,7)84 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#330, #10686115, #10686124, … |
| maximal normal subgroup of | Dic3×#330, #10686115, #10686124, … |
| maximal quotient of | Dic3×#387, Dic3×#389, Dic3×#382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15598);