Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#2020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
342 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,18,23)(14,24,28,35)(15,29,33,38)(16,22,17,21)(19,27,36,26)(20,32,39,31)(25,40,42,34)(30,37,41,43);(9,11)(12,14)(13,19)(15,30)(16,27)(17,26)(18,28)(20,37)(21,35)(22,24)(23,36)(25,32)(29,40)(31,42)(33,41)(34,38)(39,43);(12,15,18,33)(13,20,23,39)(14,25,28,42)(16,32,17,31)(19,34,36,40)(21,38,22,29)(24,37,35,43)(26,30,27,41)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5255);