Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | Q8⋊C27 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | (C22⋊C27)⋊C2 |
| commutator, hypocenter | Q8⋊C27 | C4 |
| Frattini | C22×C9 | S4 |
| Fitting | C2×Q8×C9 | D3 |
| socle, prime radical | C22×C3 | (C22⋊C9)⋊C2 |
| cosocle | C2×Q8⋊C27 | C2 |
| LCS | G ▷ Q8⋊C27 | |
| UCS | C1 ◁ C22 | |
62 conjugacy classes of subgroups.
| transitive degree | μt(G) = 864 |
Polycyclic presentation:
2^2,3^3,2^3|a^2=b,c^3=d,d^3=e,f^2=h,g^2=h,c^a=c^2d^2e^2,d^a=d^2e^2,e^a=e^2,f^a=gh,g^a=fh,f^c=g,g^c=fgh,g^f=gh72 conjugacy classes — view the full character table.
| maximal subgroup of | #13136124, C4×#88390, #13136126, … |
| maximal normal subgroup of | #13136124, C4×#88390, #13136126, … |
| maximal quotient of | (Q8⋊C27)⋊C8, C2×#11720108 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 75);