Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#1929 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D4 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C22×C7 | C4×D4 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C7×C8 ◁ G | |
224 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(9,10)(12,13,16,20,17,15,24,19)(14,18,22,26,23,21,27,25);(8,9,11,10)(12,14,16,22,17,23,24,27)(13,18,20,26,15,21,19,25);(8,10,11,9)(12,15)(13,17)(14,21)(16,19)(18,23)(20,24)(22,25)(26,27)266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5164);