Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#1868 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #120 |
| commutator | C23 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #385 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
260 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 260 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)(8,9,12,16,13,17,23,29)(10,14,20,26,19,25,33,37)(11,15,22,28,21,27,34,38)(18,30,32,39,31,24,35,36);(8,10,12,20,13,19,23,33)(9,14,16,26,17,25,29,37)(11,18,22,32,21,31,34,35)(15,24,28,36,27,30,38,39);(8,11)(9,15)(10,18)(12,22)(13,21)(14,24)(16,28)(17,27)(19,31)(20,32)(23,34)(25,30)(26,36)(29,38)(33,35)(37,39)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5103);