Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×#1271 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C22⋊C4 |
| commutator | C2×C4 | C2×C3×C8 |
| Frattini, cosocle | C2×C42 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
158 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 158 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) = 384 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,5,4,6,7,8)(9,10)(11,15,16,13)(12,14)(17,18,20,22,21,24,19,23)(25,26,27);(9,11,12,16)(10,13,14,15)(18,22)(19,20)(23,24)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 407);