Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#2166 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #117 |
| commutator | C2×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C8 | C23×C7 |
| socle | C2×C7 | #432 |
| prime radical | C23×C7 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C7×#97 ◁ G | |
122 conjugacy classes of subgroups.
| 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)(8,9,11,17,12,18,25,34,13,16,24,32,26,15,29,31)(10,19,21,33,22,30,36,39,20,14,23,28,35,27,37,38);(8,10)(9,14)(11,23)(12,22)(13,20)(15,30)(16,19)(17,33)(18,27)(21,24)(25,37)(26,35)(28,32)(29,36)(31,38)(34,39);(9,15,16,18)(10,20)(11,24)(14,27,19,30)(17,34,32,31)(22,35)(25,29)(28,39,33,38)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5401);