Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×#546 ⋊ C2 |
| Sylow | C3 ⋊ D4×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#546 | C2 |
| hypercenter | #546 | D3 |
| hypocenter | C3 | D4×#41 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #546 | |
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)(8,10,11,14)(9,13,12,15);(9,12)(13,15);(5,7);(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^e=h^275 conjugacy classes — view the full character table.
| maximal subgroup of | #11670256, #11670260, #11670262, … |
| maximal normal subgroup of | #11670256, #11670260, #11670262, … |
| maximal quotient of | C22⋊C4×#237, #10954387, #10954500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19202);