Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×C4×#41 ⋊ C2 |
| Sylow | C3 ⋊ #3498 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×C4×#41 | C2 |
| hypercenter | C4×#41 | D3 |
| hypocenter | C3 | #3498 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×#41 | |
1748 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1748 conjugacy classes of subgroups — too many to draw. Only its 557 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)(8,10,11,14)(9,13,12,15);(9,12)(13,15);(4,5,6,7);(8,9)(10,13)(11,12)(14,15);(2,3)(5,7)(9,12)(10,14)Polycyclic presentation:
2^7,3|c^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,h^a=h^2,d^b=df84 conjugacy classes — view the full character table.
| maximal subgroup of | #10687095, #10687099, #10687403, … |
| maximal normal subgroup of | #10687095, #10687099, #10687403, … |
| maximal quotient of | #10955291, #10955564, #10955592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18422);