Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2221 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C23×C7 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×#131 ◁ G | |
148 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 148 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)(9,15)(10,21)(11,16)(12,13)(17,32)(18,29)(19,28)(20,30)(22,36)(23,26)(24,25)(27,33)(34,39)(35,38);(8,9,16,29,13,19,25,30,14,20,24,28,12,18,11,15)(10,17,31,32,21,33,36,38,23,34,37,39,26,35,22,27);(8,10)(9,17)(11,22)(12,26)(13,21)(14,23)(15,27)(16,31)(18,35)(19,33)(20,34)(24,37)(25,36)(28,39)(29,32)(30,38)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5456);