Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#1302 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22⋊C4 |
| commutator | C2×C4 | C42×C7 |
| Frattini, cosocle | C2×C42 | C22×C7 |
| socle | C22×C7 | #95 |
| prime radical | C23×C7 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
144 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 144 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) = 31 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17,19,23,21,24,20,29)(18,25,27,31,26,22,28,30);(9,12,14,13)(16,18,20,28,21,26,19,27)(17,22,23,30,24,25,29,31)266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 434);