Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | Dic3×2−1+4 |
| Sylow | C3 ⋊ C4×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator | C2×C3 | C24×C4 |
| Fitting, prime radical | C2×C3×2−1+4 | C2 |
| socle, cosocle | C22×C3 | C25 |
| hypercenter | C2×2−1+4 | D3 |
| hypocenter | C3 | C4×2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×2−1+4 | |
1492 conjugacy classes of subgroups; only the 1063 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 1063 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) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(6,12)(10,13)(11,20)(18,19)(21,22,23);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20);(1,2,3,4)(22,23)102 conjugacy classes — view the full character table.
| maximal subgroup of | #10801078, Dic3×#454, #10801089, … |
| maximal normal subgroup of | #10801078, Dic3×#454, #10801089, … |
| maximal quotient of | Dic3×#520, Dic3×#519, Dic3×#536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19865);