Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4625⋊C2 |
| automorphism | Aut(G) = #4625⋊C2 (order 384) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #4625⋊C2 |
| unambiguous semidirect product | (#1020⋊C2)⋊C2 |
| unambiguous semidirect product | #5450⋊C2 |
| Fitting | H(3,𝔽4) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | #120⋊C3 | C22 |
| Frattini | C23 | C2×S4 |
| Fitting | H(3,𝔽4) | D3 |
| socle | C22 | GL(2,ℤ/4) |
| hypocenter | C22⋊A4 | D4 |
| LCS | G ▷ #120⋊C3 ▷ C22⋊A4 | |
| UCS | C1 | |
123 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 123 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:
(1,2,16,5,7,15)(3,13,10)(4,9,14,12,6,8);(2,3)(4,12)(6,13)(7,11)(9,10)(14,15)Polycyclic presentation:
2^3,3,2^4|b^a=bc,d^a=d^2,e^a=f,f^a=e,g^a=gh,e^b=eg,f^b=fh,e^c=egh,f^c=fg,e^d=f,f^d=ef,g^d=h,h^d=gh16 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5678);