Finite Group DatabasebetaFinite Group Databasebeta| order | 1323 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C72 ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | ΓL(1,𝔽8)2 |
| commutator, Fitting, socle, cosocle | C3×C72 | C32 |
| prime radical | C3×#3861 | C3 |
| hypocenter | C72 | 3−1+2 |
| LCS | G ▷ C3×C72 ▷ C72 | |
| UCS | C1 ◁ C3 | |
44 conjugacy classes of subgroups.
| transitive degree | μt(G) = 63 |
Polycyclic presentation:
3^3,7^2|a^3=c,b^3=c^2,b^a=bc,e^a=e^2,d^b=d^259 conjugacy classes — view the full character table.
| maximal subgroup of | #72224635, #72224641, #72229749, … |
| maximal normal subgroup of | #72224635, #72224641, #72229749, … |
| maximal quotient of | C2×#13050505 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1323, 25);