Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #66560⋊C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ D8 |
| Sylow | C7 ⋊ C3×#213 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4245⋊C3 |
| commutator, Fitting, cosocle | C3×C4×C7 | C22×C3 |
| Frattini, hypercenter | C4 | D3×AGL(1,𝔽7) |
| socle | C2×C3×C7 | C3×D4 |
| hypocenter | C3×C7 | C3×D8 |
| LCS | G ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
104 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^2,3,2^2,3,7|d^2=e,b^a=bd,d^a=de,g^a=g^6,d^b=de,f^b=f^2,g^c=g^245 conjugacy classes — view the full character table.
| maximal subgroup of | #72245539, AGL(1,𝔽7)×#213, #72245547, … |
| maximal normal subgroup of | #72245539, AGL(1,𝔽7)×#213, #72245547, … |
| maximal quotient of | #72254226, C3×#66560⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 223);