Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#2008 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
380 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 88 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)(10,18)(11,16)(12,15)(13,17)(14,19)(20,23)(21,22);(8,10)(9,14)(11,21)(12,19)(13,20)(15,23)(16,18)(17,22);(9,15)(11,12)(14,18)(16,17)(19,21)(22,23)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5243);