Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C8 · Schur multiplier (H2) = C42 |
| direct product | C7×#1274 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C42 |
| commutator | C4 | C4×C7×C8 |
| Frattini, cosocle | C2×C42 | C22×C7 |
| socle | C22×C7 | #95 |
| prime radical | C23×C7 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,10,12,11,13,14,15)(16,17)(18,24,26,30)(19,22)(20,23)(21,25,28,31)(27,29);(16,18)(17,21,23,28)(19,25)(20,26)(22,24,29,30)(27,31)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 406);