Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#1953 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #121 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #389 |
| prime radical | C23×C7 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
184 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 108 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)(8,9,13,17,14,18,12,28)(10,19,21,15,23,30,22,31)(11,24,26,32,27,38,25,16)(20,29,34,37,36,39,35,33);(8,10,13,21,14,23,12,22)(9,15,17,30,18,31,28,19)(11,20,26,34,27,36,25,35)(16,29,24,37,32,39,38,33);(8,11,12,25,14,27,13,26)(9,16,17,24,18,32,28,38)(10,20,22,35,23,36,21,34)(15,29,30,37,31,39,19,33)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5188);