Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#5734 |
| Fitting | C2×C3×C4⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ C2×#555 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×C4⋊Q8 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C2×C4⋊Q8 | D3 |
| hypocenter | C3 | C2×#555 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
1240 conjugacy classes of subgroups; only the 501 normal subgroups are shown.
This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 501 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(3,4,5)(6,8,9,12)(7,10,11,13)(15,19)(18,21);(6,7,9,11)(8,13,12,10)(14,16)(15,18)(17,20)(19,21);(14,15,17,19)(16,21,20,18);(4,5)(8,12)(10,13)(16,20)(18,21);(1,2)78 conjugacy classes — view the full character table.
| maximal subgroup of | #10686957, #10687308, #10687312, … |
| maximal normal subgroup of | #10686957, #10687308, #10687312, … |
| maximal quotient of | C2×#75979, C2×#74324, C2×#75985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18776);