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) = C23 |
| direct product | C3×#2117 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C2×D4 |
| commutator | C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C2×C22⋊C8 |
| prime radical | C2×C3×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C22×C3×C8 ◁ G | |
220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 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:
(1,2,5,11,6,12,20,29,7,9,21,25,4,10,18,27)(3,13,15,30,14,23,31,28,16,26,19,32,17,8,22,24)(33,34,35);(1,3)(2,8)(4,17)(5,19)(6,14)(7,16)(9,23)(10,26)(11,28)(12,13)(15,21)(18,31)(20,22)(24,25)(27,30)(29,32);(2,9)(8,23)(10,12)(11,25)(13,26)(24,28)(27,29)(30,32)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5408);