Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×#85549 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #1020 |
| commutator, cosocle, hypocenter | #543⋊C3 | C22 |
| Frattini | C22 | C2×#1020 |
| Fitting | C2×#543 | D3 |
| prime radical | C2×#543⋊C3 | C2 |
| LCS | G ▷ #543⋊C3 | |
| UCS | C1 ◁ C23 | |
552 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
42 conjugacy classes — view the full character table.
| maximal subgroup of | #72209763, #72209935, #72227601, … |
| maximal normal subgroup of | #72209935, #72227601, C2×#11711413, … |
| maximal quotient of | #72209842, #72209856, #72209880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089955);