Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#2987 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C24 |
| commutator | C22 | C23×C4×C7 |
| Frattini, cosocle | C2×C4 | C24×C7 |
| socle | C22×C7 | C23×C4 |
| prime radical | C23×C7 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
314 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,16)(11,14)(12,15)(13,18)(17,22)(19,21)(20,23)(24,25,27,31,28,30,34,37)(26,29,33,36,32,35,38,39);(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,22,23,21)(24,26)(25,29)(27,33)(28,32)(30,35)(31,36)(34,38)(37,39);(8,11,12,20)(9,14,15,23)(10,17,18,19)(13,21,16,22)(25,30)(26,32)(31,37)(33,38);(9,15)(13,16)(14,23)(21,22)(26,32)(29,35)(33,38)(36,39)266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 17062);