Finite Group DatabasebetaFinite Group Databasebeta| order | 6174 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C7 |
| direct product | AGL(1,𝔽7)×#3862 |
| Fitting, socle, derived, Sylow | C73 ⋊ C2×C32 |
| Hall | ΓL(1,𝔽8)×#3862 ⋊ C2 |
| Hall | C72×D7 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C73 | C2×C32 |
| LCS | G ▷ C73 | |
| UCS | C1 | |
124 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 126 |
133 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)2×AGL(1,𝔽7), AGL(1,𝔽7)×#64425 |
| maximal normal subgroup of | ΓL(1,𝔽8)2×AGL(1,𝔽7), AGL(1,𝔽7)×#64425 |
Past the SmallGroups library coverage — no SmallGroups id.