Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×#870 |
| Sylow | C3 ⋊ C4×#125 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D3 |
| commutator | C22×C3 | C23×C4 |
| Frattini | C23 | C23×D3 |
| Fitting | C3×C43 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | C43 | D3 |
| hypocenter | C3 | C4×#125 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ C43 | |
484 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 229 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) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(9,13)(12,15)(16,18,19,22)(17,21,20,23);(8,10,11,14)(9,12,13,15)(17,20)(21,23);(2,3)(8,9)(10,12)(11,13)(14,15)(16,17,19,20)(18,21,22,23);(4,5,6,7)120 conjugacy classes — view the full character table.
| maximal subgroup of | #10683158, #10683232, #10683235, … |
| maximal normal subgroup of | #10683158, #10683232, #10683235, … |
| maximal quotient of | C4×#4933, C4×#4928, C4×#4936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5880);