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×C4 |
| direct product | C7×#1976 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #121 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #390 |
| prime radical | C2×C7×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
284 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 284 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)(10,14,19,28)(11,21)(12,24)(13,16)(15,22)(17,25)(18,30,31,39)(20,34,33,27)(23,35)(26,37,38,32)(29,36);(8,10)(9,14,16,28)(11,18)(12,20)(13,19)(15,26,29,38)(17,27,24,34)(21,30,35,39)(22,32)(23,31)(25,33)(36,37);(8,11,12,22,13,23,25,36)(9,15,17,21,16,29,24,35)(10,18,20,32,19,31,33,37)(14,26,27,30,28,38,34,39)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5211);