Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#1796 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #379 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C2×C7×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
536 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 132 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)(19,33,35,32)(21,31)(23,30,39,37)(26,29)(28,34,36,38);(8,10,13,21)(9,14,17,29)(11,19,24,35)(12,26,27,20)(15,28,22,36)(16,31,25,18)(23,38,39,34)(30,33,37,32);(8,11)(9,15)(10,19)(12,23)(13,24)(14,28)(16,30)(17,22)(18,32)(20,34)(21,35)(25,37)(26,38)(27,39)(29,36)(31,33)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5031);