Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C9 |
| direct product | Q8×C9⋊D9 |
| Fitting | Q8×C92 ⋊ C2 |
| Sylow | C92 ⋊ C2×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C9⋊D9 |
| commutator, cosocle | C2×C92 | C23 |
| Frattini, socle | C2×C32 | C22×C3⋊D3 |
| Fitting | Q8×C92 | C2 |
| prime radical | C2×C9⋊D9 | C22 |
| hypercenter | Q8 | C9⋊D9 |
| hypocenter | C92 | C2×Q8 |
| LCS | G ▷ C2×C92 ▷ C92 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
437 conjugacy classes of subgroups.
| transitive degree | μt(G) = 648 |
210 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×C9⋊D9, #41×C9⋊D9, C2×Q8×C9⋊D9, … |
| maximal normal subgroup of | SD16×C9⋊D9, #41×C9⋊D9, C2×Q8×C9⋊D9, … |
| maximal quotient of | C4⋊C4×C9⋊D9, C2×Q8×C9⋊D9 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1173);