Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2205 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C23×C7 |
| prime radical | C7×2−1+4 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C7×Q8 ◁ G | |
224 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)(10,18)(11,21)(12,16)(13,23)(14,19)(15,20)(17,22);(8,10,14,13)(9,15,19,21)(11,22,20,12)(16,18,17,23);(8,11,14,20)(9,16,19,17)(10,12,13,22)(15,23,21,18)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5440);