Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#2251 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C2×C7×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ C4×C7×C8 ◁ G | |
178 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 43 points:
(1,2,3,4,5,6,7)(9,11)(12,13,17,21)(14,22,24,36)(15,27,29,42)(16,30,28,20)(18,31,32,25)(19,34,35,23)(26,41,40,38)(33,39,37,43);(12,14,23,39,16,26,29,31,17,24,34,43,28,40,15,25)(13,18,27,38,20,33,35,36,21,32,42,41,30,37,19,22);(8,9,10,11)(14,24)(18,32)(22,36)(25,31)(26,40)(33,37)(38,41)(39,43)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5486);