Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C82 · Schur multiplier (H2) = C4 |
| direct product | C7×#1311 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7×C8 | C22 |
| commutator | C2 | C7×C82 |
| Frattini, cosocle | C4×C8 | C22×C7 |
| socle, prime radical | C22×C7 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7×C8 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7)(16,17,18,22,19,23,24,29,20,21,25,27,26,28,30,31);(8,9,10,12,11,13,14,15)(17,21)(22,27)(23,28)(29,31)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 443);