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×#1778 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×D4 |
| commutator | C4 | C2×C42×C7 |
| Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C22×C7 | C2×C4⋊C4 |
| prime radical | C23×C7 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
220 conjugacy classes of subgroups.
| 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)(8,9,11,10)(12,13,16,21,17,19,15,20)(14,22,24,18,25,27,23,26);(12,14,15,24,17,25,16,23)(13,18,20,27,19,26,21,22);(8,10,11,9)(13,19)(18,26)(20,21)(22,27)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5013);