Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#1812 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #117 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #380 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×M16 ◁ G | |
260 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 116 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)(11,22,24,15)(12,25)(13,17)(14,20)(16,26)(19,32,34,30)(21,31)(23,29,38,37)(27,33,36,39)(28,35);(8,10,12,20,13,21,26,35)(9,14,16,18,17,28,25,31)(11,19,23,33,24,34,38,39)(15,27,29,30,22,36,37,32);(8,11)(9,15)(10,19)(12,23)(13,24)(14,27)(16,29)(17,22)(18,30)(20,33)(21,34)(25,37)(26,38)(28,36)(31,32)(35,39)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5047);