Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #2616⋊C3 |
| Fitting, Sylow | #2616 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2616 | C3 |
| prime radical | #5948 | C2 |
| hypercenter | C4 | #120⋊C3 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 | |
171 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 171 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) = 32 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 32 points:
(3,4,5)(6,7,8)(9,19,29,17,12,27,14,26,20,11,23,32)(10,24,28,13,16,31,18,22,25,15,21,30);(1,8,3,2)(4,7,5,6)(11,17)(12,22)(13,15)(16,26)(19,21)(20,32)(23,24)(25,31)(27,29)(28,30)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=h,d^2=h,e^2=h,b^a=bh,d^a=dfh,e^a=egh,d^c=efg,e^c=degh,f^c=g,g^c=fgh,e^d=eh24 conjugacy classes — view the full character table.
| maximal subgroup of | #32716⋊C3, #32717⋊C3, #32886⋊C3, … |
| maximal normal subgroup of | #32716⋊C3, #32717⋊C3, #32886⋊C3, … |
| maximal quotient of | #14302⋊C3, #14337⋊C3, #14889⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5768);