Finite Group DatabasebetaFinite Group Databasebeta| order | 12224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C4⋊Q8×C191 ⋊ C2 |
| Sylow | C191 ⋊ #554 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D191 |
| commutator, socle, cosocle | C22×C191 | C24 |
| Fitting | C4⋊Q8×C191 | C2 |
| prime radical | C2×D764 | C22 |
| hypercenter | C4⋊Q8 | D191 |
| hypocenter | C191 | #554 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,191|a^2=e,b^2=ef,d^2=e,b^a=be,c^a=cf,g^a=g^190,d^b=de,g^b=g^190,g^c=g^1901355 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #37571589, #37571593, #37571651, … |
| maximal normal subgroup of | #37571589, #37571593, #37571651, … |
| maximal quotient of | #37575396, #37575420, #37575429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12224, 1260);