Finite Group DatabasebetaFinite Group Databasebeta| order | 12224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #124×C191 ⋊ C2 |
| Sylow | C191 ⋊ #562 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D191 |
| commutator, socle, cosocle | C22×C191 | C24 |
| Fitting | #124×C191 | C2 |
| hypercenter | #124 | D191 |
| hypocenter | C191 | #562 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ #124 | |
220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,191|c^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 | #37571811, #37571814, #37571834, … |
| maximal normal subgroup of | #37571811, #37571814, #37571834, … |
| maximal quotient of | #37575662, #37575666, #37575671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12224, 1293);