Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#5697 |
| Fitting | C2×C3×#125 ⋊ C2 |
| Sylow | C3 ⋊ C2×#540 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×#125 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C2×#125 | D3 |
| hypocenter | C3 | C2×#540 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
1260 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 485 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)(7,11,10,13)(15,19)(18,21);(6,7)(8,11)(9,10)(12,13)(14,16,17,20)(15,18,19,21);(7,10)(11,13)(14,15,17,19)(16,18,20,21);(4,5)(8,12)(11,13)(16,20)(18,21);(1,2)72 conjugacy classes — view the full character table.
| maximal subgroup of | #10687722, #10800353, #10800481, … |
| maximal normal subgroup of | #10687722, #10800353, #10800481, … |
| maximal quotient of | C2×#74224, C2×#75969, C2×#76198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18668);