Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C7×#1277 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | D4 |
| commutator | C4 | C4×C7×C8 |
| Frattini, cosocle | C2×C42 | C22×C7 |
| socle, prime radical | C23×C7 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C2×C42×C7 ◁ G | |
180 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 896 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(13,17)(14,15)(18,19)(20,21,22,24,23,25,26,27);(9,11)(12,13,15,18,16,19,14,17)392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 409);