Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C7×#2115 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | C22⋊C4 |
| commutator | C22 | C22×C7×C8 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C2×C22⋊C8 |
| prime radical | C2×C7×#41 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ G | |
220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 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)(8,9,11,16,12,17,23,31,13,15,24,28,25,29,36,38)(10,18,20,30,21,32,34,39,19,14,22,26,33,27,35,37);(8,10)(9,14)(11,22)(12,21)(13,19)(15,18)(16,30)(17,27)(20,24)(23,35)(25,33)(26,28)(29,32)(31,39)(34,36)(37,38);(10,19)(14,18)(20,22)(21,33)(26,30)(27,32)(34,35)(37,39)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5350);