Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#1886 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #120 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C23×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
324 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,3)(5,6,10,16,11,15,9,14)(7,8,18,12,17,20,13,19)(21,22,23);(1,3,4,2)(5,7)(6,12)(8,16)(9,18)(10,13)(11,17)(14,20)(15,19);(5,8,11,20)(6,13,15,18)(7,16,17,14)(9,12,10,19)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5177);