Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C2 |
| direct product | C3×#1302 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22⋊C4 |
| commutator | C2×C4 | C3×C42 |
| Frattini, cosocle | C2×C42 | C22×C3 |
| socle | C22×C3 | #95 |
| prime radical | C23×C3 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
144 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 2 generators on 27 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,16,14,17,13,22)(11,18,20,24,19,15,21,23)(25,26,27);(2,5,7,6)(9,11,13,21,14,19,12,20)(10,15,16,23,17,18,22,24)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 438);