Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of #4245⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#4246⋊C3 |
| unambiguous semidirect product | #4246⋊C32 |
| unambiguous semidirect product | (C3×#4246)⋊C3 |
| Hall | C32×ΓL(1,𝔽8) ⋊ D4 |
| Sylow | C7 ⋊ C32×#67 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×AGL(1,𝔽7) |
| commutator, cosocle | C2×C3×C7 | C22×C32 |
| Frattini | C2 | C3×D3×AGL(1,𝔽7) |
| Fitting, socle | C2×C32×C7 | C22×C3 |
| hypocenter | C3×C7 | C32×D4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2×C3 | |
272 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
99 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#67×AGL(1,𝔽7), #72222040, #72222042, … |
| maximal normal subgroup of | C3×#67×AGL(1,𝔽7), #72222040, #72222042, … |
| maximal quotient of | C3×#72222059, C2×C3×#4246⋊C3, C3×#66565⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 605);