Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#1833 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #117 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #382 |
| prime radical | C23×C7 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C7×#117 ◁ G | |
172 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18,21,14)(11,22,24,37)(12,17)(13,16)(15,23,28,38)(19,32)(20,31,35,27)(25,29)(26,33)(30,39)(34,36);(8,10,12,20,13,21,25,35)(9,14,17,27,16,18,29,31)(11,19,23,33,24,34,38,39)(15,26,22,36,28,30,37,32);(8,11)(9,15,16,28)(10,19)(12,23)(13,24)(14,26,18,30)(17,22,29,37)(20,33)(21,34)(25,38)(27,36,31,32)(35,39)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5068);