Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#3395 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C7 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C7 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C4×C7×Q8 ◁ G | |
348 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 896 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7)(9,13)(14,15)(16,17,20,23)(18,24,26,31)(19,27,29,22)(21,28,30,25);(8,9,12,13)(10,14,11,15);(8,10,12,11)(9,14,13,15)(16,18,20,26)(17,21,23,30)(19,28,29,25)(22,31,27,24);(8,11,12,10)(9,14,13,15)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 17470);