Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #1088⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ #95 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #1088⋊C2 |
| commutator | #272 | C42 |
| Frattini, socle | C22×C3 | GO+(4,𝔽2) |
| Fitting | C22×H(3,𝔽3) | D4 |
| prime radical, cosocle | C22×#272 | C22 |
| hypocenter | H(3,𝔽3) | #95 |
| LCS | G ▷ #272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C22 | |
222 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^5,3^3|a^2=e,b^2=d,b^a=bc,f^a=f^2,g^a=gh,h^a=h^2,f^b=gh^2,g^b=fh^2,h^b=h^2,f^c=f^2,g^c=g^2h^2,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #13146184, #13146563, #13146564, … |
| maximal normal subgroup of | #13146184, #13146563, #13146564, … |
| maximal quotient of | #13136697, #13136698, #13136699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 674);