Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#1913 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #386 |
| commutator | C23 | C22×C4×C7 |
| Frattini, cosocle | C24 | C23×C7 |
| prime radical | C7×#534 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C24×C7 ◁ G | |
380 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(8,9)(10,13,17,21)(11,18,19,14)(12,15)(16,20)(22,23);(8,10)(9,13,15,21)(11,16)(12,17)(14,20,18,23)(19,22);(8,11)(9,14)(10,16)(12,19)(13,20)(15,18)(17,22)(21,23)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5148);