Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Hall | C3×C7 ⋊ 2−1+4 |
| Sylow | C7 ⋊ #1007 |
| Sylow | C3 ⋊ #6554 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×D7 |
| commutator, socle, cosocle | C2×C3×C7 | C24 |
| Fitting | C3×C7×Q8 | C22 |
| hypercenter | Q8 | D3×D7 |
| hypocenter | C3×C7 | 2−1+4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
292 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
Polycyclic presentation:
2^5,3,7|c^2=e,d^2=e,b^a=be,g^a=g^6,d^b=de,f^b=f^2,d^c=de72 conjugacy classes — view the full character table.
| maximal subgroup of | #13059309, #13059315, #13059318, … |
| maximal normal subgroup of | #13059309, #13059315, #13059318, … |
| maximal quotient of | #13058532, #13058534, #13058537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1159);