Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2155 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #106 |
| commutator | C8 | C22×C4×C7 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C2×#106 |
| prime radical | C23×C7 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ G | |
144 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 100 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)(10,19,22,33)(11,16)(12,25)(13,28)(14,18)(15,21,30,35)(17,26)(20,32,23,37)(24,31)(27,38)(29,34,39,36);(8,10,12,21,13,23,26,36,14,22,27,35,11,20,24,34)(9,15,17,19,16,29,25,32,18,30,31,33,28,39,38,37);(8,11,14,13)(9,16,18,28)(10,20,22,23)(12,24,27,26)(15,29,30,39)(17,25,31,38)(19,32,33,37)(21,34,35,36)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5390);