Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#2237 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C23×C7 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ G | |
132 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 60 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)(16,17,21,26)(18,22,29,37)(19,23,32,39)(20,33,31,25)(24,40,38,30)(27,34,43,46)(28,44,42,36)(35,47,45,41);(9,12)(13,15)(16,18,21,29)(17,22,26,37)(19,27,32,43)(20,28,31,42)(23,34,39,46)(24,35,38,45)(25,36,33,44)(30,41,40,47);(8,10,11,14)(9,13,12,15)(17,23)(19,30)(20,31)(22,34)(24,25)(26,39)(27,41)(28,42)(32,40)(33,38)(35,36)(37,46)(43,47)(44,45)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5472);