Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
| direct product | SL(2,𝔽3)×(C3⋊D3)⋊C3 |
| unambiguous semidirect product | #88623⋊C3 |
| unambiguous semidirect product | (Q8×(C3⋊D3)⋊C3)⋊C3 |
| unambiguous semidirect product | (C3⋊D3×SL(2,𝔽3))⋊C3 |
| Fitting | Q8×H(3,𝔽3) ⋊ C2×C3 |
| derived | C32×Q8 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C32×Q8 | C2×C32 |
| Frattini, socle | C2×C3 | C3×D3×A4 |
| Fitting | Q8×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C2 | |
224 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | #72220370, #72220376, SL(2,𝔽3)×PSU(3,𝔽2)⋊C3, … |
| maximal normal subgroup of | #72220376, SL(2,𝔽3)×((C3⋊D3)⋊C3)⋊C2, GL(2,𝔽3)×(C3⋊D3)⋊C3, … |
| maximal quotient of | C2×SL(2,𝔽3)×(C3⋊D3)⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2153);