Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#1659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C7 |
| prime radical | C7×H(3,𝔽4) | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C42×C7 ◁ G | |
228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 224 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(9,12)(10,14)(13,15)(16,17,21,25)(18,26,28,31)(19,24,20,23)(22,29,30,27);(10,14)(12,15)(16,18)(17,22)(19,29)(20,27)(21,28)(23,31)(24,26)(25,30);(8,9)(10,15)(11,13)(12,14)(16,18)(17,22)(19,27)(20,29)(21,28)(23,26)(24,31)(25,30)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 4894);