Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C4 |
| direct product | C7×#2242 |
| 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 | |
138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,30)(19,24)(20,38)(21,41)(22,42)(23,29)(25,34)(26,31)(27,40)(28,39)(32,44)(33,43)(35,47)(36,46)(37,45);(9,12)(13,15)(16,18,31,36,22,33,40,37,23,34,21,32,39,35,20,19)(17,24,38,47,28,44,41,25,29,45,27,43,42,46,26,30);(8,10,11,14)(9,13,12,15)(16,19,20,35,39,32,21,34,23,37,40,33,22,36,31,18)(17,25,26,44,42,47,27,24,29,30,41,46,28,43,38,45)266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5477);