Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2148 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #102 |
| commutator | C8 | C22×C4×C7 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | C2×#102 |
| prime radical | C2×C7×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C22×C7×C8 ◁ G | |
220 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 100 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)(8,9,11,15,13,18,25,17)(10,19,22,33,23,36,20,35)(12,26,24,16,28,32,37,34)(14,27,30,21,31,38,29,39);(8,10)(9,14)(11,20)(12,27)(13,23)(15,29)(16,33)(17,30)(18,31)(19,32)(21,37)(22,25)(24,39)(26,36)(28,38)(34,35);(8,11,13,25)(9,15,18,17)(10,20,23,22)(12,24,28,37)(14,29,31,30)(16,32,34,26)(19,35,36,33)(21,27,39,38)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5383);