Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#3416 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C7 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C7×#41 ◁ C7×#131 ◁ G | |
360 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 180 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 — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,18)(10,15,19,29)(11,14,24,28)(12,25,23,17)(16,32,27,22)(20,26,33,37)(21,36,34,31)(30,39,38,35);(9,14)(11,22)(12,23)(15,26)(16,17)(18,28)(20,35)(21,34)(24,32)(25,27)(29,37)(30,31)(33,39)(36,38);(8,10)(9,15)(11,20)(12,21)(13,19)(14,26)(16,30)(17,31)(18,29)(22,35)(23,34)(24,33)(25,36)(27,38)(28,37)(32,39);(10,19)(15,29)(20,33)(21,34)(26,37)(30,38)(31,36)(35,39)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 17491);