Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×C3⋊#564 |
| unambiguous semidirect product | (C2×C3)⋊#564 |
| Sylow | C3 ⋊ C2×#564 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×#125 | C2 |
| socle, prime radical | C23×C3 | C24 |
| hypercenter | C2×#125 | D3 |
| hypocenter | C3 | C2×#564 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
896 conjugacy classes of subgroups; only the 469 normal subgroups are shown.
This group has 896 conjugacy classes of subgroups — too many to draw. Only its 469 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2)(3,4,5)(6,9,10,17)(7,12,13,19)(8,16,15,21)(11,18,14,20)(23,29)(27,35)(28,37)(34,36);(6,8)(7,11)(9,16)(10,15)(12,18)(13,14)(17,21)(19,20)(22,25,26,33)(23,28,29,37)(24,30,31,32)(27,34,35,36);(8,15)(9,17)(11,14)(12,19)(22,24,26,31)(23,27,29,35)(25,32,33,30)(28,36,37,34);(4,5)(6,7,10,13)(8,14,15,11)(9,12,17,19)(16,20,21,18)(22,23)(24,27)(25,28)(26,29)(30,34)(31,35)(32,36)(33,37);(1,2)66 conjugacy classes — view the full character table.
| maximal subgroup of | #10687641, #10687642, C2×#71250, … |
| maximal normal subgroup of | #10687641, #10687642, C2×#71250, … |
| maximal quotient of | C2×#75993, C2×#74978, C2×#74979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18664);