Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#2053 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #394 |
| prime radical | C7×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
264 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 80 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)(10,19)(11,24)(12,29)(13,17)(14,18)(15,21)(16,26)(20,30)(22,32)(23,31)(25,33)(27,34)(28,37)(35,38)(36,39);(8,10)(9,15)(11,25)(12,22)(13,21)(14,23)(16,33)(17,31)(18,32)(19,29)(20,27)(24,30)(26,36)(28,35)(34,38)(37,39);(8,11,12,26,14,28,13,27)(9,16,17,24,18,34,29,37)(10,20,21,35,23,36,22,25)(15,30,19,33,32,39,31,38)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5288);