Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #2825⋊C3 |
| Fitting, Sylow | #2825 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #560⋊C3 |
| commutator, cosocle, hypocenter | #95 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2825 | C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C2 | |
122 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 20 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:
(1,3,20,5,10,11)(2,6,19,8,9,16)(4,12,22)(7,18,23)(13,17,24)(14,15,21);(1,13)(2,14)(3,6)(4,8)(5,7)(9,10)(11,23)(12,18)(15,17)(16,21)(19,22)(20,24)Polycyclic presentation:
2^2,3,2^5|a^2=f,d^2=gh,e^2=h,c^a=cg,d^a=dfgh,e^a=efh,c^b=cg,d^b=dfh,e^b=eg,d^c=ef,e^c=deg,g^c=fgh,h^c=fg,e^d=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2825⋊C3, #36676⋊C3, #52398⋊C3, … |
| maximal normal subgroup of | C2×#2825⋊C3, #36676⋊C3, #52398⋊C3, … |
| maximal quotient of | #14637⋊C3, #14651⋊C3, #14656⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5783);