Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C2×#99 |
| commutator | C23 | C22×C4×C7 |
| Frattini, cosocle | C2×D4 | C23×C7 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 84 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)(10,18)(11,15)(12,25,26,16)(13,17)(14,20,28,35)(19,32)(21,33)(22,36)(23,38,39,30)(24,31)(27,34,29,37);(8,10,13,21)(9,14,17,28)(11,22,24,19)(12,20,26,35)(15,29,31,27)(16,18,25,33)(23,37,39,34)(30,36,38,32);(8,11,13,24)(9,15,17,31)(10,19,21,22)(12,23,26,39)(14,27,28,29)(16,30,25,38)(18,32,33,36)(20,34,35,37)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5358);