Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2214 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C4×C7×C8 ◁ G | |
150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 150 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) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,15,21)(10,19,24,30)(11,25,27,38)(12,20,14,18)(13,16,23,31)(17,32,34,22)(26,39,29,35)(28,36,37,33);(8,10,14,23,15,24,12,13)(9,16,20,30,21,31,18,19)(11,22,29,37,27,32,26,28)(17,25,36,39,34,38,33,35);(8,11)(9,17)(10,22)(12,26)(13,28)(14,29)(15,27)(16,25)(18,33)(19,35)(20,36)(21,34)(23,37)(24,32)(30,39)(31,38)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5449);