Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×#3482⋊C3 |
| unambiguous semidirect product | (C2×#3482)⋊C3 |
| Fitting, Sylow | C2×#3482 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×A4 |
| commutator, cosocle | C2×Q8 | C24×C3 |
| Frattini | C22 | C24×A4 |
| Fitting | C2×#3482 | C3 |
| socle | C23 | C23×A4 |
| hypercenter | C2×C4×D4 | A4 |
| hypocenter | Q8 | C2×C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×D4 | |
1776 conjugacy classes of subgroups; only the 277 normal subgroups are shown.
This group has 1776 conjugacy classes of subgroups — too many to draw. Only its 277 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085074);