Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#85467 |
| Fitting | C2×#576 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×S4 |
| commutator, cosocle | #41⋊C3 | C24 |
| Frattini | C4 | C23×S4 |
| Fitting | C2×#576 | D3 |
| hypercenter | C2×Q16 | S4 |
| hypocenter | SL(2,𝔽3) | C22×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×Q16 | |
1366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 1366 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11923454, #72226229, C2×#11715001, … |
| maximal normal subgroup of | #72226229, C2×#11715001 |
| maximal quotient of | #72226229, D3×#85467 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089359);