Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#343 ⋊ C2 |
| Sylow | C3 ⋊ #2006 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #882 |
| commutator, cosocle | C22×C3×C4 | C23 |
| Frattini | C22×C4 | C22×D3 |
| Fitting | C3×#343 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #343 | D3 |
| hypocenter | C3 | #2006 |
| LCS | G ▷ C22×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #343 | |
406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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 — 3 generators on 15 points:
(1,2)(5,7)(8,9)(10,13)(11,15)(12,14);(2,3)(4,5)(6,7)(8,9)(10,15)(11,13)(12,14);(2,3)(9,13)(10,11)(14,15)Polycyclic presentation:
2^7,3|d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=h^2,c^b=cf,d^b=dg,h^b=h^2,e^c=eg,h^c=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10707251, #10707256, #10707269, … |
| maximal normal subgroup of | #10707251, #10707256, #10707269, … |
| maximal quotient of | #10695498, #10695505, #10695512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1584);