Finite Group DatabasebetaFinite Group Databasebeta| order | 7648 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Dic239⋊C8 |
| Sylow | C239 ⋊ C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×D239 |
| commutator | C2×C239 | C2×C8 |
| Fitting | C2×C8×C239 | C2 |
| socle | C22×C239 | C2×C4 |
| hypercenter | C2×C8 | D239 |
| hypocenter | C239 | C4⋊C8 |
| LCS | G ▷ C2×C239 ▷ C239 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8 | |
38 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,239|a^2=d,b^2=c,d^2=e,b^a=bc,f^b=f^2381924 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4⋊C8×D239, C2×Dic239⋊C8, #29504794, … |
| maximal normal subgroup of | C4⋊C8×D239, C2×Dic239⋊C8, #29504794, … |
| maximal quotient of | #29504522, #29504524, #29504526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7648, 75);