Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#2230 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #476 |
| 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 | |
126 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 126 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) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(9,12)(10,18,20,15)(11,13)(16,17)(19,21,22,23);(8,9,11,12,14,17,13,16)(10,15,19,21,20,18,22,23);(8,10)(9,15)(11,19)(12,21)(13,22)(14,20)(16,23)(17,18)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5465);