Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#1763 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×D4 |
| commutator | C4 | C2×C3×C42 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
324 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 384 |
A minimal faithful permutation representation — 3 generators on 19 points:
(5,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,18,19);(1,2,3,4)(9,11,12,15,14,10,13,16);(5,7,8,6)(9,12,14,13)(10,15,11,16)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5054);