Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#2170 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C16 | D4 |
| commutator | C4 | C22×C7×C8 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C8×D4 |
| prime radical | C7×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C16 ◁ C2×C7×C16 ◁ G | |
146 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 92 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)(8,9,13,17,12,18,27,35,14,19,29,36,28,16,32,34)(10,20,22,25,23,31,37,26,24,33,38,39,21,15,30,11);(8,10)(9,15)(11,17)(12,23)(13,22)(14,24)(16,33)(18,20)(19,31)(21,28)(25,35)(26,36)(27,37)(29,38)(30,32)(34,39);(8,11,13,20,12,25,27,31,14,26,29,33,28,39,32,15)(9,10,17,22,18,23,35,37,19,24,36,38,16,21,34,30)392 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5405);