Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C82 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#1273 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C7 | C22 |
| commutator | C2 | C7×C82 |
| Frattini, cosocle | C2×C42 | C22×C7 |
| socle, prime radical | C23×C7 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C7 ◁ G | |
212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,14,16,15,17,18,19);(9,11)(20,21,22,24,23,25,26,27)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 405);