Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #560⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (#126⋊C3)⋊C4 |
| unambiguous semidirect product | #2383⋊C3 |
| Fitting, Sylow | #2383 ⋊ C3 |
| derived | C42 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #560⋊C3 |
| commutator, hypocenter | C42 | C2×C3×C4 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | #2383 | C3 |
| prime radical | C2×#126⋊C3 | C2 |
| cosocle | C2×C42 | C22×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C2 | |
116 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,8,7)(9,20,10,11,18,16)(12,15,17,14,13,19);(1,2,3,4)(5,19,7,17)(6,18,8,20)(9,10,13,14)(11,12,15,16)Polycyclic presentation:
2^2,3,2^5|a^2=dg,e^2=g,f^2=h,c^a=cg,e^a=eh,f^a=fgh,c^b=cgh,e^b=egh,f^b=fg,d^c=dh,e^c=efg,f^c=eg,g^c=gh,h^c=g34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×(#126⋊C3)⋊C4, C4×#560⋊C3, #45970⋊C3, … |
| maximal normal subgroup of | C2×(#126⋊C3)⋊C4, C4×#560⋊C3, #45970⋊C3, … |
| maximal quotient of | #13830⋊C3, #14562⋊C3, #14620⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5756);