Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C22×C42 |
| direct product | C7×#2641 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C7 |
| prime radical | C2×C7×#127 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
584 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 200 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,17)(15,16)(18,19)(21,24)(25,26);(8,9,10,11)(13,16)(17,18)(20,21,23,24)(22,25,27,26);(8,9)(10,11)(12,13)(14,18)(15,16)(17,19)(20,21)(22,26)(23,24)(25,27);(12,14,15,19)(13,17,16,18)(20,22,23,27)(21,25,24,26)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 16716);