Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #5450 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #2375⋊C3 |
| Fitting, Sylow | #2375 ⋊ C3 |
| derived | C24 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5450 |
| commutator, hypocenter | C24 | C2×C3×C4 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2375 | C3 |
| prime radical | C2×#120⋊C3 | C2 |
| cosocle | C25 | C22×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2 | |
158 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,7)(6,10)(8,20,9,15,16,14)(11,17,12,19,13,18);(1,2,3,4)(5,12,8,6)(7,19,15,17)(9,11,16,13)(10,14,18,20)Polycyclic presentation:
2^2,3,2^5|a^2=d,e^a=eh,f^a=fgh,e^b=egh,f^b=fg,e^c=efh,f^c=eg,g^c=gh,h^c=g34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2375⋊C3, C4×#5450, #45439⋊C3, … |
| maximal normal subgroup of | C2×#2375⋊C3, C4×#5450, #45439⋊C3, … |
| maximal quotient of | #11709220, #14601⋊C3, #14609⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5754);