Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C7×M16 ⋊ C2 |
| Hall | C3×C7 ⋊ #131 |
| Sylow | C7 ⋊ #907 |
| Sylow | C3 ⋊ #6470 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×D21 |
| commutator, socle | C2×C3×C7 | C22×C4 |
| Fitting | C3×C7×M16 | C2 |
| prime radical | #66728 | C2 |
| cosocle | C3×C4×C7 | C23 |
| hypercenter | M16 | D21 |
| hypocenter | C3×C7 | #131 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C4 ◁ M16 | |
124 conjugacy classes of subgroups.
| transitive degree | μt(G) = 336 |
Polycyclic presentation:
2^5,3,7|c^2=d,d^2=e,b^a=be,f^a=f^2,g^a=g^6,c^b=ce120 conjugacy classes — view the full character table.
| maximal subgroup of | #13053564, #13053571, #13053572, … |
| maximal normal subgroup of | #13053564, #13053571, #13053572, … |
| maximal quotient of | #13056339, #13056345, #13056346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 972);