Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C33 |
No aliases for this group.
| Fitting | C32×#41 ⋊ D3 |
| derived | C32×SL(2,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊(C3⋊S4) |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C22 |
| Fitting | C32×#41 | D3 |
| socle | C2×C32 | C2×S4 |
| hypercenter | C4 | C3⋊(C3⋊S4) |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
460 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^2,3^3,2^3|b^2=h,f^2=h,g^2=h,b^a=bh,c^a=c^2,d^a=d^2,e^a=e^2,f^a=gh,g^a=fh,f^c=g,g^c=fgh,g^f=gh69 conjugacy classes — view the full character table.
| maximal subgroup of | #13165974, #13165975, #13165980, … |
| maximal normal subgroup of | #13165974, #13165975, #13165980, … |
| maximal quotient of | #13173041, C4⋊(C3⋊(C3⋊GL(2,𝔽3))), #13173044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4438);