Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#1829 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #381 |
| commutator | C23 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C22×C7×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
272 conjugacy classes of subgroups.
| 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)(8,9,11,15,12,13,19,20)(10,16,18,23,17,14,21,22);(9,13)(11,19)(14,16)(18,21);(8,10)(9,14,13,16)(11,18)(12,17)(15,22,20,23)(19,21)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5064);