Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#2165 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #117 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | #432 |
| prime radical | C23×C7 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C7×#97 ◁ G | |
128 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)(8,9,10,15,11,16,18,23,12,13,17,20,19,14,21,22);(9,13)(10,17)(14,16)(18,21);(9,14,13,16)(10,17)(15,23,20,22)(18,21)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5400);