Finite Group DatabasebetaFinite Group Databasebeta| order | 12224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Fitting | #120×C191 ⋊ C2 |
| Sylow | C191 ⋊ H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D191 |
| commutator, socle, cosocle | C22×C191 | C24 |
| Fitting | #120×C191 | C2 |
| hypercenter | #120 | D191 |
| hypocenter | C191 | H(3,𝔽4) |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ #120 | |
260 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,191|a^2=e,b^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de,g^b=g^1901349 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #37574519, #37574522 |
| maximal normal subgroup of | #37574519, #37574522 |
| maximal quotient of | #37575658, #37575659, #37575661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12224, 1292);