Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#413 ⋊ C2 |
| Sylow | C3 ⋊ #3068 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#884 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#413 | C2 |
| socle | C2×C3 | C2×#121 |
| hypercenter | #413 | D3 |
| hypocenter | C3 | #3068 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #413 | |
466 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,11)(9,17)(10,18)(16,19);(2,3)(4,6)(5,10)(7,13)(8,14)(9,16)(11,18)(12,15)(17,19);(2,3)(4,5,7,9,8,11,15,17)(6,10,13,16,14,18,12,19);(2,3)(5,9)(6,12)(7,15)(11,17)(13,14)(16,19)Polycyclic presentation:
2^7,3|b^2=e,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^2,d^b=dg,h^b=h^2,f^c=fg,h^c=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | #10832699, #10832706, #10832721, … |
| maximal normal subgroup of | #10832699, #10832706, #10832721, … |
| maximal quotient of | #10777958, #10777991, #10778002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11296);