Finite Group DatabasebetaFinite Group Databasebeta| order | 12096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | AGL(1,𝔽7)×A4⋊S4 |
| Fitting, socle | C24×C7 ⋊ C2×C3×C3⋊D3 |
| derived | C7×A42 ⋊ C22×C3 |
| Sylow | C7 ⋊ C2×C3×A4⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×A42 | C22×C3 |
| Fitting, socle | C24×C7 | C2×C3×C3⋊D3 |
| LCS | G ▷ C7×A42 | |
| UCS | C1 | |
1516 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 1516 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
98 conjugacy classes — view the full character table.
| maximal subgroup of | S42×AGL(1,𝔽7), C2×AGL(1,𝔽7)×A4⋊S4 |
| maximal normal subgroup of | S42×AGL(1,𝔽7), C2×AGL(1,𝔽7)×A4⋊S4 |
| maximal quotient of | C2×AGL(1,𝔽7)×A4⋊S4, AGL(1,𝔽7)×SO+(4,𝔽3) |
| holomorph of | C7×A4 |
Past the SmallGroups library coverage — no SmallGroups id.