Finite Group DatabasebetaFinite Group Databasebeta| order | 2352 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | AGL(1,𝔽7)×D28 |
| Fitting | C4×C72 ⋊ C22×C3 |
| Hall | D7×D28 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C2×D4 |
| Sylow | C72 ⋊ C2×C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C72 | C23×C3 |
| Fitting | C4×C72 | C22×C3 |
| hypercenter | C4 | D7×AGL(1,𝔽7) |
| hypocenter | C72 | C2×C3×D4 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ C4 | |
224 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
119 conjugacy classes — view the full character table.
| maximal subgroup of | #72245780, #72245804, D4×D7×AGL(1,𝔽7), … |
| maximal normal subgroup of | #72245780, #72245804, D4×D7×AGL(1,𝔽7), … |
| maximal quotient of | #72245905 |
Past the SmallGroups library coverage — no SmallGroups id.