Finite Group DatabasebetaFinite Group Databasebeta| order | 12224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | #120×D191 |
| Fitting | #120×C191 ⋊ C2 |
| Sylow | C191 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D191 |
| commutator, socle, cosocle | C22×C191 | C24 |
| Fitting | #120×C191 | C2 |
| hypercenter | #120 | D191 |
| hypocenter | C191 | C2×#120 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ #120 | |
662 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1358 conjugacy classes. The character table is too large to display.
| maximal subgroup of | AΣL(1,𝔽16)×D191, B(4,𝔽2)×D191 |
| maximal normal subgroup of | AΣL(1,𝔽16)×D191, B(4,𝔽2)×D191 |
| maximal quotient of | #379×D191, #385×D191, #386×D191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12224, 1033);