Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#533 ⋊ C2 |
| Sylow | C3 ⋊ Q8×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C3×#533 | C2 |
| prime radical | C2×#235 | C22 |
| hypercenter | #533 | D3 |
| hypocenter | C3 | Q8×#41 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #533 | |
1068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(12,14,15,18)(13,17,16,19);(13,16)(17,19);(4,6,7,11)(5,8,9,10);(2,3)(12,13)(14,17)(15,16)(18,19);(2,3)(4,5,7,9)(6,10,11,8)(13,16)(17,19)Polycyclic presentation:
2^7,3|a^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,h^a=h^2,d^b=df,h^b=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10687227, #10687231, #10762063, … |
| maximal normal subgroup of | #10687227, #10687231, #10762063, … |
| maximal quotient of | #10954498, #10954501, #10954609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18853);