Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#1909 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #386 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
312 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 108 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)(16,27)(19,32,35,39)(21,33)(23,30,26,31)(28,34,38,36)(29,37);(8,10,12,20,13,21,27,37)(9,14,16,18,17,29,25,33)(11,19,26,36,24,35,23,34)(15,28,31,32,22,38,30,39);(8,11)(9,15)(10,19)(12,26)(13,24)(14,28)(16,31)(17,22)(18,32)(20,36)(21,35)(23,27)(25,30)(29,38)(33,39)(34,37)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5144);