Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2047 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C23×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 256 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) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,17,21)(14,22,25,37)(15,26,29,19)(16,30,32,20)(18,31,34,24)(23,39,27,38)(28,36,43,41)(33,40,35,42);(12,14,17,25)(13,18,21,34)(15,27,29,23)(16,31,32,24)(19,35,26,33)(20,37,30,22)(28,40,43,42)(36,38,41,39);(9,11)(12,15,16,28,17,29,32,43)(13,19,20,36,21,26,30,41)(14,23,24,40,25,27,31,42)(18,33,22,38,34,35,37,39)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5282);