Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#2070 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C23×C7 |
| prime radical | C23×C7 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
196 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)(8,9,12,26,13,17,28,16)(10,18,20,14,21,34,38,30)(11,22,24,31,25,15,27,32)(19,35,36,29,23,33,37,39);(8,10,12,20,13,21,28,38)(9,14,16,18,17,30,26,34)(11,23,24,37,25,19,27,36)(15,29,31,35,22,39,32,33);(8,11,13,25)(9,15,17,22)(10,19,21,23)(12,27,28,24)(14,29,30,39)(16,32,26,31)(18,33,34,35)(20,37,38,36)161 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 5305);