Finite Group DatabasebetaFinite Group Databasebeta| order | 12224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #123×C191 ⋊ C2 |
| Sylow | C191 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D191 |
| commutator, socle, cosocle | C22×C191 | C24 |
| Fitting | #123×C191 | C2 |
| prime radical | C2×#15690286 | C2 |
| hypercenter | #123 | D191 |
| hypocenter | C191 | #536 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ #123 | |
270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,191|b^2=e,c^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce,g^b=g^1901352 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #37574172, #37581924 |
| maximal normal subgroup of | #37574172, #37581924 |
| maximal quotient of | #37575105, #37575107, #37575139, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12224, 1121);