Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#6553 |
| Fitting | C3×C7×#41 ⋊ C2 |
| Hall | C3×C7 ⋊ H(5,𝔽2) |
| Sylow | C7 ⋊ C3×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C23×D7 |
| commutator, cosocle | C2×C7 | C24×C3 |
| Frattini | C2 | C23×C3×D7 |
| Fitting | C3×C7×#41 | C2 |
| socle | C2×C3×C7 | C24 |
| hypercenter | C3×#41 | D7 |
| hypocenter | C7 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C3 ◁ C3×#41 | |
332 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2,3)(4,5,7,9,10,8,6)(11,13,14,17)(12,16,15,18);(12,15)(16,18);(11,12)(13,16)(14,15)(17,18);(5,6)(7,8)(9,10)(13,17)(16,18)141 conjugacy classes — view the full character table.
| maximal subgroup of | #13054797, #13054831, #13054839, … |
| maximal normal subgroup of | #13054797, #13054831, #13054839, … |
| maximal quotient of | C3×#90223, C3×#90210, C3×#90212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1214);