Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4656 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
| direct product | C2×#5223 |
| Fitting | C23×C3×D4 ⋊ C2 |
| Sylow | C3 ⋊ C2×#392 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C23×C3×D4 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | C2×#392 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×D4 | |
1812 conjugacy classes of subgroups; only the 285 normal subgroups are shown.
This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 285 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(3,4,5)(6,7)(8,9)(11,13)(14,15)(16,17);(1,2)(6,7)(8,9)(10,11)(12,13)(15,17);(1,2)(4,5)(7,9)(10,11)(12,13);(1,2)84 conjugacy classes — view the full character table.
| maximal subgroup of | #10690193, #10690208, #10695380, … |
| maximal normal subgroup of | #10690193, #10690208, #10695380, … |
| maximal quotient of | C2×#71550, C2×#72131, C2×#71570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16627);