Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C7×Q16 ⋊ C2 |
| Hall | C3×C7 ⋊ #135 |
| Sylow | C7 ⋊ #919 |
| Sylow | C3 ⋊ #6482 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D21 |
| commutator, cosocle | C3×C4×C7 | C23 |
| Frattini | C4 | C22×D21 |
| Fitting | C3×C7×Q16 | C2 |
| socle | C2×C3×C7 | C2×D4 |
| hypercenter | Q16 | D21 |
| hypocenter | C3×C7 | #135 |
| LCS | G ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
124 conjugacy classes of subgroups.
| transitive degree | μt(G) = 336 |
Polycyclic presentation:
2^5,3,7|b^2=e,c^2=e,d^2=e,b^a=be,c^a=ce,f^a=f^2,g^a=g^6,c^b=cd,d^b=de,d^c=de84 conjugacy classes — view the full character table.
| maximal subgroup of | #13053672, #13053678, #13053687, … |
| maximal normal subgroup of | #13053672, #13053678, #13053687, … |
| maximal quotient of | #13056428, #13056432, #13056436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 984);