Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C22×#944 |
| Sylow | C3 ⋊ C22×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C23×C3×Q8 | C2 |
| socle, prime radical | C24×C3 | C23 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | C22×C4⋊Q8 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×Q8 | |
1624 conjugacy classes of subgroups; only the 725 normal subgroups are shown.
This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 725 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) = 384 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(3,4)(5,6,7)(9,11)(12,14,15,19)(13,16,17,18);(12,13,15,17)(14,18,19,16);(6,7)(8,9,10,11);(3,4);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | #10687819, #10701737, #10701738, … |
| maximal normal subgroup of | #10687819, #10701737, #10701738, … |
| maximal quotient of | C22×#4975, C22×#4976, C22×#4973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19822);