Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#1958 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
336 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 128 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 — 3 generators on 27 points:
(1,2,4,3)(6,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,23)(22,24)(25,26,27);(1,3,4,2)(9,11,12,18,13,19,21,24)(10,14,16,17,15,22,20,23);(5,6,7,8)(10,15)(14,22)(16,20)(17,23)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5249);