Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C32 |
| direct product | C2×#72213251 |
| Fitting | C25×C32 ⋊ C3⋊D3 |
| derived | (C22⋊A4)⋊H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊(A4⋊S4) |
| commutator, cosocle, hypocenter | (C22⋊A4)⋊H(3,𝔽3) | C22 |
| Frattini | C3 | C2×C3⋊(A4⋊S4) |
| Fitting | C25×C32 | C3⋊D3 |
| socle | C25×C3 | C3⋊(C3⋊D3) |
| LCS | G ▷ (C22⋊A4)⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
2069 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 2069 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
100 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72213328, C22×#72213251 |
| maximal normal subgroup of | C2×#72213328, C22×#72213251 |
| maximal quotient of | C2×#72220369, C22×#72213251 |
Past the SmallGroups library coverage — no SmallGroups id.