Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | H(5,𝔽2) ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | A42 |
| commutator, Fitting, cosocle | C3×H(5,𝔽2) | C32 |
| prime radical | C3×#997 | C3 |
| hypocenter | H(5,𝔽2) | 3−1+2 |
| LCS | G ▷ C3×H(5,𝔽2) ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C3 | |
82 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
3^3,2^5|a^3=c,b^3=c,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bc,f^a=fg,g^a=f,d^b=de,e^b=d,e^d=eh,g^f=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #13164056, #13164060, C2×#11722286, … |
| maximal normal subgroup of | #13164056, #13164060, C2×#11722286, … |
| maximal quotient of | #13146418, C2×#11722286, #72244963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2253);