Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#1735 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | D4 |
| commutator | C4 | C2×C42×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C23×C7 | C2×D4 |
| prime radical | C24×C7 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C23×C4×C7 ◁ G | |
424 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,14,15)(16,17)(18,26)(19,23)(20,24)(21,25)(22,27)(28,31)(29,30);(9,11)(16,18,20,28,21,29,19,27)(17,22,23,30,25,31,24,26);(16,19,21,20)(17,23,25,24)(18,27,29,28)(22,30,31,26)392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4970);