Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#485 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C11 |
| prime radical | C23×C11 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,21,24,28)(22,25,30,34)(23,26,31,27)(29,32,35,33);(13,16)(17,19)(20,22,24,30)(21,25,28,34)(23,29,31,35)(26,32,27,33);(12,14,15,18)(13,17,16,19)(21,26)(23,31)(25,32)(27,28)(29,35)(33,34)209 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 891);