Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #2643⋊C3 |
| Fitting, Sylow | #2643 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#126⋊C3 |
| commutator, cosocle, hypocenter | #95 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2643 | C3 |
| hypercenter | C22 | #126⋊C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C2 ◁ C22 | |
112 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 19 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:
(1,3,9,5,12,20)(2,15,16)(4,10,22,14,11,21)(6,7,19)(8,18,24)(13,17,23);(1,8)(2,4)(3,15)(5,13)(6,14)(7,12)(9,23,20,24)(10,18)(11,17)(16,21,19,22)Polycyclic presentation:
2^2,3,2^5|d^2=g,e^2=gh,b^a=bf,c^a=cfg,d^a=dfgh,e^a=eh,c^b=cfg,d^c=eh,e^c=deg,g^c=gh,h^c=fg,e^d=ef24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2643⋊C3, #34639⋊C3, #35914⋊C3, … |
| maximal normal subgroup of | C2×#2643⋊C3, #34639⋊C3, #35914⋊C3, … |
| maximal quotient of | #14331⋊C3, #14333⋊C3, #14349⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5769);