Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | D3×C7×#67 |
| Fitting | C22×C32×C7 ⋊ C22 |
| Hall | C32×C7 ⋊ C2×D4 |
| Sylow | C32 ⋊ C2×C7×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C2×D32 |
| commutator, cosocle | C2×C32 | C23×C7 |
| Frattini | C2 | C2×D32×C7 |
| Fitting | C22×C32×C7 | C22 |
| socle | C2×C32×C7 | C23 |
| hypercenter | C22×C7 | D32 |
| hypocenter | C32 | C2×C7×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 | |
232 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
189 conjugacy classes — view the full character table.
| maximal subgroup of | D32×C7×D4, D3×D7×#67, D3×#66694, … |
| maximal normal subgroup of | D32×C7×D4, D3×D7×#67, D3×#66694, … |
| maximal quotient of | C7×#63977, C7×#63983 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 769);