Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#88447 |
| derived, Sylow | H(3,𝔽3) ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, hypocenter | H(3,𝔽3) | C2×C42 |
| Frattini | C22×C3 | C2×D32 |
| Fitting, prime radical | C23×H(3,𝔽3) | C22 |
| socle | C23×C3 | D32 |
| cosocle | C22×H(3,𝔽3) | C23 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C23 | |
432 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
88 conjugacy classes — view the full character table.
| maximal subgroup of | #13136720, C4×#88447, #13140055, … |
| maximal normal subgroup of | #13136720, C4×#88447, #13140055, … |
| maximal quotient of | C2×#11720302, #13139921, #13139922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1262);