Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C9 |
No aliases for this group.
| Fitting | Q8×C92 ⋊ C2 |
| Sylow | C92 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C9⋊D9 |
| commutator, cosocle | C2×C92 | C23 |
| Frattini, socle | C2×C32 | C22×C3⋊D3 |
| Fitting | Q8×C92 | C2 |
| hypercenter | Q8 | C9⋊D9 |
| hypocenter | C92 | #41 |
| LCS | G ▷ C2×C92 ▷ C92 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
460 conjugacy classes of subgroups.
| transitive degree | μt(G) = 648 |
Polycyclic presentation:
2^4,3^4|b^2=d,c^2=d,e^3=h,f^3=g,c^a=cd,e^a=e^2h^2,f^a=f^2g^2,g^a=g^2,h^a=h^2,c^b=cd210 conjugacy classes — view the full character table.
| maximal subgroup of | #72244189, #41×C9⋊D9, C2×#13047456 |
| maximal normal subgroup of | #72244189, #41×C9⋊D9, C2×#13047456 |
| maximal quotient of | #72244173, C2×#13047456 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1174);