Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C2×#99 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×Q8 | C23×C7 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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)(8,9,13,18,14,19,11,16)(10,20,23,38,24,35,21,37)(12,26,25,36,28,17,29,34)(15,30,31,27,32,39,33,22);(8,10,14,24)(9,15,19,32)(11,23,13,21)(12,27,28,22)(16,33,18,31)(17,35,26,20)(25,30,29,39)(34,37,36,38);(8,11,14,13)(9,16,19,18)(10,21,24,23)(12,25,28,29)(15,31,32,33)(17,34,26,36)(20,37,35,38)(22,30,27,39)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5370);