Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×#532 ⋊ C2 |
| Sylow | C3 ⋊ D4×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C3×#532 | C2 |
| hypercenter | #532 | D3 |
| hypocenter | C3 | D4×#41 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #532 | |
1648 conjugacy classes of subgroups.
| 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);(2,3)(5,7);(2,3)(8,9)(10,13)(11,12)(14,15);(2,3)(4,5)(6,7)(9,12)(13,15)Polycyclic presentation:
2^7,3|e^2=f,b^a=bf,c^a=cg,h^a=h^2,d^b=df,h^b=h^2,h^c=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10687206, #10687210, #10762045, … |
| maximal normal subgroup of | #10687206, #10687210, #10762045, … |
| maximal quotient of | #10954388, #10954498, #10954588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18814);