Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C2×#99 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×Q8 | C23×C7 |
| prime radical | C2×C7×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C23×C7 ◁ C22×C7×Q8 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,16,13,17,28,35)(10,18,21,36,19,32,26,38)(11,22,24,34,25,15,27,33)(14,29,31,23,30,37,39,20);(8,10)(9,14)(11,23)(12,26)(13,19)(15,32)(16,31)(17,30)(18,22)(20,25)(21,28)(24,29)(27,37)(33,38)(34,36)(35,39);(10,19)(14,30)(18,32)(20,23)(21,26)(29,37)(31,39)(36,38)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5368);