Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#1773 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D4 |
| commutator | C4 | C2×C42×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C22×C7 | C4×D4 |
| prime radical | C23×C7 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C7×C8 ◁ G | |
224 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 152 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)(13,17,19,22)(14,20)(18,25,26,24)(21,27);(8,9,10,11)(12,13)(14,18)(15,22)(16,19)(17,23)(20,26)(21,25)(24,27);(12,14,15,21,16,20,23,27)(13,18,22,25,19,26,17,24)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5008);