Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#2231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| prime radical | C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C4×C7×C8 ◁ G | |
120 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 120 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) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,14,19)(10,20,23,18)(11,24)(12,28,21,17)(13,15,30,32)(16,33)(22,36)(25,38)(26,37)(27,31)(29,34)(35,39);(8,10,21,30,14,23,12,13)(9,15,28,18,19,32,17,20)(11,22,26,39,27,33,25,29)(16,31,35,37,36,24,34,38);(8,11)(9,16)(10,22)(12,25)(13,29)(14,27)(15,31)(17,34)(18,37)(19,36)(20,38)(21,26)(23,33)(24,32)(28,35)(30,39)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5466);