Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | Q8⋊H(3,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3×C4 | C3⋊S4 |
| commutator, hypocenter | Q8⋊H(3,𝔽3) | C4 |
| Fitting | C32×#41 | D3 |
| socle | C2×C3 | C3⋊SL(2,ℤ/4) |
| prime radical, cosocle | #41⋊H(3,𝔽3) | C2 |
| LCS | G ▷ Q8⋊H(3,𝔽3) | |
| UCS | C1 ◁ C3×C4 | |
122 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^2,3^3,2^3|a^2=b,b^2=h,f^2=h,g^2=h,c^a=c^2,d^a=d^2,f^a=gh,g^a=fh,d^c=de,f^c=g,g^c=fgh,g^f=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #13146373, #13146375, #13146376, … |
| maximal normal subgroup of | #13146373, #13146375, #13146376, … |
| maximal quotient of | #13136786, C2×#11720740 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 707);