Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C22×#882 |
| Fitting | C22×C3×C22⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ C22×#120 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C22×C3×C22⋊C4 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C22×C22⋊C4 | D3 |
| hypocenter | C3 | C22×#120 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×C22⋊C4 | |
4544 conjugacy classes of subgroups; only the 725 normal subgroups are shown.
This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 725 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2)(3,4)(5,6,7)(9,11)(12,14,15,13);(1,2)(3,4)(8,9)(10,11)(12,13)(14,15);(6,7)(12,13)(14,15);(3,4);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | #10687806, #10701678, #10701797, … |
| maximal normal subgroup of | #10687806, #10701678, #10701797, … |
| maximal quotient of | C22×#4955, C22×#4668, C22×#4950, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18267);