Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×#73048 |
| Fitting | C23×C42 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#857 |
| commutator, cosocle | C2×C42⋊C3 | C23 |
| Frattini | C23 | C22×S4 |
| Fitting | C23×C42 | D3 |
| socle | C24 | C2×S4 |
| hypercenter | C23 | #857 |
| hypocenter | C42⋊C3 | C2×D4 |
| LCS | G ▷ C2×C42⋊C3 ▷ C42⋊C3 | |
| UCS | C1 ◁ C22 ◁ C23 | |
854 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 854 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11712792, C2×#72213079, #72218177, … |
| maximal normal subgroup of | C2×#11712792, #72218177, #72226107, … |
| maximal quotient of | C2×#11711328, #72226086 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087130);