Finite Group DatabasebetaFinite Group Databasebeta| order | 12224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C191⋊#564 |
| Sylow | C191 ⋊ #564 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D191 |
| commutator, socle, prime radical, cosocle | C22×C191 | C24 |
| Fitting | #125×C191 | C2 |
| hypercenter | #125 | D191 |
| hypocenter | C191 | #564 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ #125 | |
172 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,191|a^2=e,b^2=f,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de,g^d=g^1901349 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #37571821, #37571823, #37571846, … |
| maximal normal subgroup of | #37571821, #37571823, #37571846, … |
| maximal quotient of | #37575702, #37575703, #37575704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12224, 1304);