Finite Group DatabasebetaFinite Group Databasebeta| order | 22112 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q16×C691 ⋊ C2 |
| Sylow | C691 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D691 |
| commutator, cosocle | C4×C691 | C23 |
| Frattini | C4 | C22×D691 |
| Fitting | Q16×C691 | C2 |
| socle | C2×C691 | C2×D4 |
| hypercenter | Q16 | D691 |
| hypocenter | C691 | #135 |
| LCS | G ▷ C4×C691 ▷ C2×C691 ▷ C691 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
62 conjugacy classes of subgroups.
| transitive degree | μt(G) = 11056 |
Polycyclic presentation:
2^5,691|c^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^690,d^b=de,f^b=f^690,f^c=f^6902429 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22112, 169);