Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D336 |
| Fitting | C3×C7×C16 ⋊ C2 |
| Hall | C3×C7 ⋊ D16 |
| Sylow | C7 ⋊ D48 |
| Sylow | C3 ⋊ D112 |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19)(20,21,22,23,24,25,26);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,26)(22,25)(23,24)Polycyclic presentation:
2^5,3,7|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^6171 conjugacy classes — view the full character table.
| maximal subgroup of | #13050847, #13050848, #13050852, … |
| maximal normal subgroup of | #13050847, #13050848, #13050852, … |
| maximal quotient of | #13051352, D672, Dic336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 220);