Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2038 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ G | |
278 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 278 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) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(9,10)(12,13)(14,17)(15,18)(16,20)(19,24)(21,25)(22,27)(23,26);(8,9)(10,11)(12,14,16,22)(13,17,20,27)(15,23,24,21)(18,26,19,25);(8,10,11,9)(13,18)(15,24)(17,25)(19,20)(21,23)(26,27)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5273);