Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×#2171 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C2×D4 |
| commutator | C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C8×D4 |
| prime radical | C3×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C2×C3×C16 ◁ G | |
144 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,6,11,5,12,22,15,7,13,24,30,23,10,27,29)(3,14,16,4,17,20,31,19,18,28,32,21,9,8,25,26)(33,34,35);(1,3)(2,8)(4,15)(5,17)(6,16)(7,18)(9,23)(10,28)(11,26)(12,14)(13,20)(19,30)(21,29)(22,31)(24,32)(25,27);(1,4,6,20,5,19,22,28,7,21,24,8,23,26,27,14)(2,9,11,25,12,3,15,16,13,17,30,31,10,18,29,32)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5462);