Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#1858 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #120 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #385 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
252 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 252 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)(8,9,13,17,14,18,12,25)(10,19,21,15,22,30,26,29)(11,16,23,32,24,33,27,31)(20,35,38,39,37,34,36,28);(8,10,13,21,14,22,12,26)(9,15,17,30,18,29,25,19)(11,20,23,38,24,37,27,36)(16,28,32,35,33,39,31,34);(8,11)(9,16)(10,20)(12,27)(13,23)(14,24)(15,28)(17,32)(18,33)(19,34)(21,38)(22,37)(25,31)(26,36)(29,39)(30,35)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5093);