Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×#1278 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | D4 |
| commutator | C4 | C3×C4×C8 |
| Frattini, cosocle | C2×C42 | C22×C3 |
| socle | C22×C3 | #95 |
| prime radical | C23×C3 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ C2×C3×C42 ◁ G | |
172 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 172 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) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,4,7,6,8,5,3)(9,10)(11,13)(12,15)(14,16)(17,18,19);(1,3,5,8,6,7,4,2)(10,13,15,14)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 414);