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) = C2 |
| direct product | C3×#1574 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C42 | C23×C3 |
| socle | C22×C3 | C2×C4⋊C4 |
| prime radical | C23×C3 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
158 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 158 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) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,4,7)(3,5,8,6)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24)(25,26,27);(1,2,3,6,4,7,8,5)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(1,3,4,8)(2,5,7,6)(11,17)(14,20)(18,23)(21,24)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4865);