Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C92 ⋊ C2×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D92 |
| commutator, cosocle | C2×C92 | C23 |
| Frattini, socle | C2×C32 | C2×D32 |
| Fitting | C4×C92 | C22 |
| prime radical | C2×C9⋊D9 | C22 |
| hypercenter | C4 | D92 |
| hypocenter | C92 | C2×Q8 |
| LCS | G ▷ C2×C92 ▷ C92 | |
| UCS | C1 ◁ C2 ◁ C4 | |
227 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3^4|a^2=d,b^2=d,c^2=d,e^3=g,f^3=h,c^a=cd,e^a=e^2g^2,g^a=g^2,c^b=cd,f^b=f^2h^2,h^b=h^2114 conjugacy classes — view the full character table.
| maximal subgroup of | #72219874, #72219899, #72219926, … |
| maximal normal subgroup of | #72219874, #72219899, #72219926, … |
| maximal quotient of | C2×#13047076 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 794);