Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #2801⋊C3 |
| Fitting, Sylow | #2801 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #560⋊C3 |
| commutator, cosocle, hypocenter | #95 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2801 | C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C2 | |
105 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 105 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,3,20)(2,9,18,6,7,16)(4,12,21,14,15,22)(5,10,11)(8,17,24)(13,19,23);(1,8,5,13)(2,4,6,14)(3,19,10,17)(7,15,9,12)(11,16)(18,20)(21,24)(22,23)Polycyclic presentation:
2^2,3,2^5|a^2=f,d^2=fg,e^2=gh,b^a=bf,c^a=cfgh,d^a=dfgh,e^a=efh,c^b=cfgh,d^b=dfh,e^b=eg,d^c=defgh,e^c=dh,g^c=fh,h^c=gh,e^d=ef24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2801⋊C3, #52299⋊C3, #52472⋊C3, … |
| maximal normal subgroup of | C2×#2801⋊C3, #52299⋊C3, #52472⋊C3, … |
| maximal quotient of | #14562⋊C3, #14618⋊C3, #14620⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5778);