Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of (C3×A4)⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22×C32 |
| direct product | C2×(C3×A4)⋊A4 |
| unambiguous semidirect product | (C2×C3×A4)⋊A4 |
| Sylow | C25 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | A42 |
| commutator, cosocle | C24×C3 | C2×C32 |
| Frattini | C3 | C2×A42 |
| Fitting, socle | C25×C3 | C32 |
| hypocenter | C24 | C2×H(3,𝔽3) |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C2×C3 | |
276 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | (C3⋊SL(2,ℤ/4))⋊A4, #13164063, #13164081, … |
| maximal normal subgroup of | (C3⋊SL(2,ℤ/4))⋊A4, #13164063, #13164081, … |
| maximal quotient of | C4×(C3×A4)⋊A4, C2×(C3×A4)⋊SL(2,𝔽3), #13164279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4003);