Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C2×#87536 |
| Fitting, Sylow | C2×H(7,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×A4 |
| commutator, cosocle, hypocenter | Q8 | C25×C3 |
| Frattini | C2 | C25×A4 |
| Fitting | C2×H(7,𝔽2) | C3 |
| hypercenter | C2×2−1+4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22 ◁ C2×2−1+4 | |
11380 conjugacy classes of subgroups; only the 1125 normal subgroups are shown.
This group has 11380 conjugacy classes of subgroups — too many to draw. Only its 1125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
134 conjugacy classes — view the full character table.
| maximal subgroup of | C2×(2−1+4⋊C3)⋊A4, C2×#11714723, #72226219, … |
| maximal normal subgroup of | C2×(2−1+4⋊C3)⋊A4, C2×#11714723, #72226219, … |
| maximal quotient of | #72226247, #72228552 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085313);