Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#2249 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C2×C7×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ C4×C7×C8 ◁ G | |
184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 184 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)(14,22)(15,27)(16,19)(17,30)(18,24)(20,23)(21,28)(25,34)(26,41)(29,42)(31,40)(32,43)(33,38)(35,37)(36,39);(12,14,23,37,17,26,29,40,16,25,38,43,28,39,15,24)(13,18,27,36,21,32,33,34,19,31,42,41,30,35,20,22);(8,9,10,11)(13,19)(18,31)(20,33)(21,30)(22,34)(27,42)(32,35)(36,41)266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5484);