Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C32 |
| direct product | C22×C3×(C3⋊D3)⋊C3 |
| Fitting | C22×C3×H(3,𝔽3) ⋊ C2 |
| derived | C32 ⋊ C23×C32 |
| Sylow | C3×H(3,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | (C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C32 | C23×C32 |
| Frattini | C3 | C22×C32×D3 |
| Fitting | C22×C3×H(3,𝔽3) | C2 |
| socle | C22×C32 | C3×D3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C3 | |
410 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | #88803⋊C3, C3×#88450, #13047721, … |
| maximal normal subgroup of | #88803⋊C3, C3×#88450, #13047721, … |
| maximal quotient of | C2×C3×#634⋊C3, C2×C3×C4×(C3⋊D3)⋊C3, C2×C3×C4⋊((C3⋊D3)⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 624);