Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×AGL(1,ℤ/16) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #442 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C3×#509 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#97 ◁ G | |
154 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,6,8,15)(3,10)(4,11)(7,13,16,9)(12,14)(17,18,19);(1,2)(3,7)(4,13)(5,8)(6,14)(9,11)(10,15)(12,16);(1,3,11,14,5,12,4,10)(2,7,9,6,8,16,13,15)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5472);