Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C25 |
| direct product | C7×#1869 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C23 |
| commutator | C22 | C22×C7×C8 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C23×C7 | C22×C4 |
| prime radical | C22×C7×D4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7)(9,11)(12,13)(14,15)(16,17,19,22,20,18,21,23);(8,9,10,11)(16,18,19,23,20,17,21,22);(13,15)392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5104);