Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#3135 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C22×D4 |
| prime radical | C3×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
392 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,22)(13,18)(14,23)(15,19)(16,20)(21,24)(25,26,27);(1,3,4,7)(2,5,6,8)(9,11,14,20,15,21,13,12)(10,16,18,17,19,22,23,24);(2,6)(5,8)(9,12,14,11,15,20,13,21)(10,17,18,22,19,24,23,16);(9,13,15,14)(10,18,19,23)(11,12,21,20)(16,17,22,24)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17575);