Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #120⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #1779⋊C3 |
| Fitting, Sylow | #1779 ⋊ C3 |
| derived | C24 ⋊ C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #120⋊C3 |
| commutator, hypocenter | C24 | C3×C8 |
| Frattini | C22×C4 | C2×A4 |
| Fitting | #1779 | C3 |
| socle | C23 | C4×A4 |
| prime radical | C2×C22⋊A4 | C4 |
| cosocle | C24×C4 | C2×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C4 | |
172 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,5,4,6,7,8)(9,10)(11,14,13,16,12,15);(10,16)(14,15)Polycyclic presentation:
2,3,2^6|a^2=c,c^2=d,e^a=eg,f^a=fh,e^b=fgh,f^b=efh,g^b=h,h^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1779⋊C3, #20579⋊C3, C8×#120⋊C3, … |
| maximal normal subgroup of | C2×#1779⋊C3, #20579⋊C3, C8×#120⋊C3, … |
| maximal quotient of | #8316⋊C3, #11380⋊C3, #12679⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 600);