Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting | C32×Q8 ⋊ Dic3 |
| derived | C32×SL(2,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4⋊#157 |
| commutator, hypocenter | C32×SL(2,𝔽3) | C4 |
| Fitting | C32×Q8 | Dic3 |
| socle | C2×C32 | SL(2,ℤ/4) |
| prime radical, cosocle | C3⋊D3×SL(2,𝔽3) | C2 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^2,3^3,2^3|a^2=b,f^2=h,g^2=h,c^a=c^2,d^a=de,e^a=de^2,f^a=gh,g^a=fh,d^b=d^2,e^b=e^2,f^c=g,g^c=fgh,g^f=gh30 conjugacy classes — view the full character table.
| maximal subgroup of | #13180174, #13180175, #13180176, … |
| maximal normal subgroup of | #13180174, #13180175, #13180176, … |
| maximal quotient of | SL(2,𝔽3)⋊#622, C2×#11724156 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4123);