Finite Group DatabasebetaFinite Group Databasebeta| order | 688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | D4×C43 ⋊ C2 |
| Sylow | C43 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D43 |
| commutator, socle, cosocle | C2×C43 | C23 |
| Fitting | D4×C43 | C2 |
| hypercenter | D4 | D43 |
| hypocenter | C43 | #41 |
| LCS | G ▷ C2×C43 ▷ C43 | |
| UCS | C1 ◁ C2 ◁ D4 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 344 |
Polycyclic presentation:
2^4,43|b^2=d,c^a=cd,e^a=e^42,c^b=cd115 conjugacy classes — view the full character table.
| maximal subgroup of | #13063344, #13063346, #13063349, … |
| maximal normal subgroup of | #13063344, #13063346, #13063349, … |
| maximal quotient of | #13063281, #13063283, (C22⋊D43)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(688, 32);