Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2186 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ G | |
230 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(13,18)(14,17)(15,22)(19,27)(20,26)(21,24)(23,25);(12,13)(14,19)(15,23)(16,20)(17,25)(18,22)(21,26)(24,27);(9,11)(12,14,16,17)(13,19,20,25)(15,21,24,22)(18,23,26,27)203 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5421);