Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C4 |
| direct product | C3×#1922 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C22×C3 | C4×D4 |
| prime radical | C23×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C8 ◁ G | |
186 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,12,15,17,13,22)(11,16,20,18,21,23,19,24)(25,26,27);(2,5)(6,8)(9,11,14,20,15,21,13,19)(10,16,12,18,17,23,22,24);(1,3,4,7)(2,6,5,8)(9,12)(10,13)(11,18)(14,17)(15,22)(16,19)(20,23)(21,24)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5213);