Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2143 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×D8 |
| 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)(8,9,12,17,14,18,13,28)(10,19,22,36,20,35,29,33)(11,23,25,34,27,38,26,16)(15,24,31,37,30,39,32,21);(8,10)(9,15)(11,24)(12,22)(13,29)(14,20)(16,33)(17,31)(18,30)(19,23)(21,26)(25,37)(27,39)(28,32)(34,36)(35,38);(10,20)(15,30)(19,35)(21,37)(22,29)(24,39)(31,32)(33,36)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5378);