Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #392⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #2397⋊C3 |
| Fitting, Sylow | #2397 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #127⋊C3 |
| commutator, cosocle, hypocenter | #95 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2397 | C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C4 | |
175 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 21 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:
(1,9,22,3,13,21,5,19,11,10,4,17)(2,14,20,6,12,24,8,18,16,15,7,23);(2,8)(4,12)(6,15)(7,13)(9,18)(14,19)Polycyclic presentation:
2^2,3,2^5|a^2=fg,b^2=fg,d^2=f,e^2=h,c^a=cfgh,d^a=dg,e^a=efgh,c^b=cfgh,d^c=efg,e^c=degh,f^c=h,g^c=fgh,h^c=gh,e^d=efg36 conjugacy classes — view the full character table.
| maximal subgroup of | #21683⋊C3, #21684⋊C3, #23358⋊C3, … |
| maximal normal subgroup of | #21683⋊C3, #21684⋊C3, #23358⋊C3, … |
| maximal quotient of | #13856⋊C3, C4×#392⋊C3, C4×#399⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5763);