Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4626⋊C2 |
| automorphism | Aut(G) = #4626⋊C2 (order 384) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #4626⋊C2 |
| unambiguous semidirect product | (#857⋊C2)⋊C2 |
| unambiguous semidirect product | (#560⋊C3)⋊C2 |
| Fitting | #560 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | #127⋊C3 | C22 |
| Frattini | C23 | C2×S4 |
| Fitting | #560 | D3 |
| socle | C22 | GL(2,ℤ/4) |
| hypocenter | C42⋊C3 | D4 |
| LCS | G ▷ #127⋊C3 ▷ C42⋊C3 | |
| UCS | C1 | |
117 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 117 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,16,5,8,10)(4,7,14,13,11,15)(6,9,12);(2,6)(3,11)(4,13)(7,12)(8,9)(14,16)Polycyclic presentation:
2^3,3,2^4|e^2=g,f^2=gh,b^a=bc,d^a=d^2,e^a=fh,f^a=eh,g^a=gh,e^b=egh,f^b=fh,e^c=eg,f^c=fgh,e^d=fg,f^d=ef,g^d=gh,h^d=g16 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5677);