Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#459 ⋊ C2 |
| Sylow | C3 ⋊ D4×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#459 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #459 | D3 |
| hypocenter | C3 | D4×D8 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
620 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 133 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 — 4 generators on 15 points:
(1,2,3)(8,10)(11,12)(13,15);(2,3)(5,7);(2,3)(8,9)(10,13)(11,15)(12,14);(2,3)(4,5)(6,7)(9,13)(10,11)(14,15)Polycyclic presentation:
2^7,3|e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=h^2,d^b=de,e^b=eg,h^b=h^2,h^c=h^2,e^d=eg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10784527, #10784529, #10784531, … |
| maximal normal subgroup of | #10784527, #10784529, #10784531, … |
| maximal quotient of | #10784531, #10784541, #10784546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12926);