Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #3518⋊C3 |
| Fitting, Sylow | #3518 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Fitting | #3518 | C3 |
| hypercenter | #124 | A4 |
| hypocenter | Q8 | C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ #124 | |
265 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,3,4,7)(2,6,5,8)(9,11,12,10)(13,18,22,15,14,24,19,16,21,17,20,23);(2,5)(6,8)(9,23)(10,22)(11,21)(12,24)(13,18)(14,15)(16,19)(17,20);(1,2)(3,6)(4,5)(7,8)(10,11)(15,17)(16,18)(23,24)Polycyclic presentation:
2^3,3,2^4|b^2=h,c^2=e,f^2=h,g^2=h,b^a=be,c^a=ch,f^d=g,g^d=fg,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #17654⋊C3, #62105⋊C3, #62106⋊C3, … |
| maximal normal subgroup of | #17654⋊C3, #62105⋊C3, #62106⋊C3, … |
| maximal quotient of | #34193⋊C3, #34262⋊C3, #34293⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18194);