Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#1664 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C7 |
| prime radical | C2×C7×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ G | |
184 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,17)(10,18,20,23)(11,16,12,15)(14,21,22,19)(24,25,29,27)(26,32,35,39)(28,36,38,31)(30,37,33,34);(8,10)(9,14)(11,21)(12,19)(13,20)(15,23)(16,18)(17,22)(24,26,29,35)(25,30,27,33)(28,37,38,34)(31,39,36,32);(8,10)(9,14)(11,19)(12,21)(13,20)(15,18)(16,23)(17,22)(24,27,29,25)(26,33,35,30)(28,31,38,36)(32,37,39,34)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4899);