Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#1742 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22⋊C4 |
| commutator | C22 | C2×C42×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C22×C7 | C2×C22⋊C4 |
| prime radical | C24×C7 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
388 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 216 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)(12,13,16,21,17,20,15,19)(14,22,23,27,24,18,25,26);(8,9,10,11)(12,14)(13,18)(15,23)(16,25)(17,24)(19,26)(20,22)(21,27);(12,15,17,16)(13,19,20,21)(14,23,24,25)(18,26,22,27)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4977);