Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2098 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| socle | C2×C7 | #399 |
| prime radical | C2×C7×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 210 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)(8,9,12,17,13,18,14,28)(10,19,21,31,23,35,22,15)(11,24,26,34,27,16,29,33)(20,36,25,39,37,32,38,30);(8,10,14,21,13,23,12,22)(9,15,28,19,18,31,17,35)(11,25,29,37,27,38,26,20)(16,32,34,30,24,36,33,39);(8,11)(9,16)(10,20)(12,26)(13,27)(14,29)(15,30)(17,33)(18,24)(19,36)(21,25)(22,38)(23,37)(28,34)(31,39)(32,35)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5333);