Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2194 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C7×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×#131 ◁ G | |
216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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)(8,9,14,20)(10,21,24,37)(11,25,29,36)(12,19,13,18)(15,31,34,22)(16,30,35,23)(17,27,32,28)(26,38,39,33);(8,10)(9,15)(11,26)(12,23)(13,30)(14,24)(16,32)(17,35)(18,33)(19,38)(20,34)(21,25)(22,28)(27,31)(29,39)(36,37);(8,10)(9,16)(11,22)(12,30)(13,23)(14,24)(15,32)(17,34)(18,37)(19,21)(20,35)(25,38)(26,28)(27,39)(29,31)(33,36)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5429);