Finite Group DatabasebetaFinite Group Databasebeta| order | 6720 |
| isoclinism | isoclinic class of AGL(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C22 |
| direct product | C5×AGL(3,𝔽2) |
| Fitting, socle | C23×C5 ⋊ GL(3,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | AGL(3,𝔽2) |
| commutator, hypocenter | AGL(3,𝔽2) | C5 |
| Frattini | C1 | G |
| Fitting, socle | C23×C5 | GL(3,𝔽2) |
| cosocle | C23 | C5×GL(3,𝔽2) |
| LCS | G ▷ AGL(3,𝔽2) | |
| UCS | C1 ◁ C5 | |
190 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
55 conjugacy classes — view the full character table.
| derived subgroup of | C2×D5×AGL(3,𝔽2) |
| maximal subgroup of | D5×AGL(3,𝔽2), C2×C5×AGL(3,𝔽2) |
| maximal normal subgroup of | D5×AGL(3,𝔽2), C2×C5×AGL(3,𝔽2) |
| maximal quotient of | C2×C5×AGL(3,𝔽2) |
Past the SmallGroups library coverage — no SmallGroups id.