Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#6461 |
| Fitting | C3×C7×C4⋊C4 ⋊ C2 |
| Hall | C3×C7 ⋊ #126 |
| Sylow | C7 ⋊ C3×#126 |
120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(4,5,7,9,10,8,6)(12,16)(15,18)(19,21,22,25)(20,24,23,26);(11,13,14,17)(12,15,16,18)(20,23)(21,25);(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,24)(22,23)(25,26)132 conjugacy classes — view the full character table.
| maximal subgroup of | #13058685, #13058686, #13058688, … |
| maximal normal subgroup of | #13058685, #13058686, #13058688, … |
| maximal quotient of | C3×#89423, C3×C7⋊#401, C3×#89432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 791);