Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2040 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C7×#537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
290 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(9,10)(12,13)(14,22)(15,20)(16,19)(17,21)(18,24)(23,27)(25,26);(8,9)(10,11)(12,14)(13,18)(15,24)(16,25)(17,23)(19,27)(20,22)(21,26);(8,10,11,9)(13,19)(14,23)(16,17)(20,21)(24,25)(26,27)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5275);