Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×#85557 |
| Fitting | C2×H(3,𝔽4) ⋊ D3 |
| derived | #5465 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #5934 |
| commutator, cosocle, hypocenter | #5465 | C22 |
| Frattini | C22 | C2×#1020 |
| Fitting | C2×H(3,𝔽4) | D3 |
| hypercenter | C23 | #1020 |
| LCS | G ▷ #5465 | |
| UCS | C1 ◁ C22 ◁ C23 | |
590 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 590 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
32 conjugacy classes — view the full character table.
| maximal subgroup of | #72209773, C2×#72209772, C2×#72209596, … |
| maximal normal subgroup of | #72227615, #72228025, #72241408, … |
| maximal quotient of | #72209838, #72209863, #72209885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090132);