Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#2129 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C2×#99 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×D4 | C23×C7 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
238 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 84 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)(10,18,21,33)(11,15)(12,25,27,16)(13,17)(14,20)(19,34,22,36)(23,38,39,31)(24,32)(26,29)(28,35)(30,37);(8,10,13,21)(9,14,17,29)(11,22,24,19)(12,26,27,20)(15,30,32,28)(16,33,25,18)(23,35,39,37)(31,34,38,36);(8,11,13,24)(9,15,17,32)(10,19,21,22)(12,23,27,39)(14,28,29,30)(16,31,25,38)(18,34,33,36)(20,35,26,37)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5364);