Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #2614⋊C3 |
| Fitting, Sylow | #2614 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2614 | C3 |
| hypercenter | C22 | #120⋊C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2 ◁ C22 | |
172 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 172 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) = 48 |
A minimal faithful permutation representation — 2 generators on 24 points:
(3,5,4)(6,7,8)(10,14)(11,17,20)(12,19,18)(15,24,22,21,23,16);(1,8,3,2)(4,6,5,7)(9,24,13,23)(10,18,14,17)(11,22,19,16)(12,15,20,21)Polycyclic presentation:
2^2,3,2^5|d^2=h,e^2=h,b^a=bh,d^a=df,e^a=eg,d^c=defh,e^c=dfgh,f^c=fg,g^c=f,e^d=eh24 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2614⋊C3, #34632⋊C3, #36078⋊C3, … |
| maximal normal subgroup of | C2×#2614⋊C3, #34632⋊C3, #36078⋊C3, … |
| maximal quotient of | #14301⋊C3, #14302⋊C3, #14336⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5766);