Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#1382 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | D8 |
| commutator | C8 | C42×C7 |
| Frattini, cosocle | C22×C8 | C22×C7 |
| socle | C22×C7 | #102 |
| prime radical | C23×C7 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ G | |
132 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,23)(15,28)(16,31)(17,33)(18,22)(19,24)(20,30)(21,29)(25,35)(26,43)(27,39)(32,36)(34,42)(37,41)(38,40);(12,14,16,25,29,40,30,41,18,27,15,24,17,26,32,42)(13,19,20,23,33,37,31,43,22,35,36,39,21,34,28,38)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 514);