Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#2009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C7×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
348 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,20)(11,24)(12,18)(13,17)(14,19)(15,22)(16,26)(21,29)(23,34)(25,33)(27,32)(28,30)(31,37)(35,36)(38,39);(8,10)(9,15)(11,21)(12,28)(13,22)(14,23)(16,31)(17,34)(18,20)(19,30)(24,35)(25,38)(26,36)(27,37)(29,33)(32,39);(8,11,14,27)(9,16,19,33)(10,21,23,37)(12,26,13,25)(15,29,30,31)(17,24,18,32)(20,35,34,39)(22,38,28,36)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5244);