Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#535 ⋊ C2 |
| Sylow | C3 ⋊ #2002 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #953 |
| commutator, cosocle | C22×C3×C4 | C23 |
| Frattini | C22×C4 | C22×D3 |
| Fitting | C3×#535 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #535 | D3 |
| hypocenter | C3 | #2002 |
| LCS | G ▷ C22×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #535 | |
478 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 478 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) = 48 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(5,7)(10,15)(12,13);(2,3)(9,12)(10,15)(13,14);(4,5)(6,7)(8,9)(10,12)(11,14)(13,15)Polycyclic presentation:
2^7,3|d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fg,c^b=cf,d^b=dg,h^b=h^2,d^c=dg48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2002, C2×#71856, #10822094, … |
| maximal normal subgroup of | D3×#2002, C2×#71856, #10822094, … |
| maximal quotient of | #10695381, #10695423, #10695437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4432);