Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#2239 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C23×C7 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×M16 ◁ G | |
124 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 56 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)(8,9,13,17)(10,18,19,36)(11,22,23,39)(12,27,25,16)(14,29,30,20)(15,32,33,26)(21,34,28,37)(24,31,38,35);(11,23)(15,33)(18,36)(20,29)(22,39)(24,38)(26,32)(34,37);(8,10,12,28,13,19,25,21)(9,14,16,35,17,30,27,31)(11,24,26,29,23,38,32,20)(15,34,22,36,33,37,39,18)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5474);