Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #13049804 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, socle, Sylow | C34 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | #66472 | C22 |
| Fitting, socle, hypocenter | C34 | SD16 |
| prime radical | #10620206 | C2 |
| LCS | G ▷ #66472 ▷ C3⋊(C3⋊(C3⋊D3)) ▷ C34 | |
| UCS | C1 | |
173 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 173 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 12 |
Polycyclic presentation:
2^4,3^4|a^2=c,c^2=d,b^a=bcd,e^a=e^2f^2,f^a=ef^2,g^a=g^2h^2,h^a=g^2,c^b=cd,e^b=f,f^b=e,g^b=g^2h,e^c=f^2,f^c=e,g^c=g^2h,h^c=gh,e^d=e^2,f^d=f^2,g^d=g^2,h^d=h^218 conjugacy classes — view the full character table.
| maximal subgroup of | #72210299, #72210346, C2×#13049804, … |
| maximal normal subgroup of | #72210346, C2×#13049804, C3×#13049804 |
| maximal quotient of | #72213356, C2×#13049804, #72213750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3522);