Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#2250 |
| 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 ◁ C42×C7 ◁ C4×C7×C8 ◁ G | |
198 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 72 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)(9,11)(13,17)(14,18)(15,21)(19,23)(20,27)(22,26)(24,25);(12,13,18,25,15,20,22,23,16,19,26,27,21,24,14,17);(8,9,10,11)(13,19)(17,23)(20,24)(25,27)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5485);