Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#473 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23×C11 |
| prime radical | C11×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ G | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,18,23)(14,24,28,34)(15,29,31,37)(16,22,17,21)(19,27,35,26)(20,32,36,30)(25,39,41,33)(38,42,43,40);(12,14,17,27,18,28,16,26)(13,19,22,34,23,35,21,24)(15,25,32,42,31,41,30,40)(20,33,37,43,36,39,29,38);(12,15)(13,20)(14,25)(16,30)(17,32)(18,31)(19,33)(21,29)(22,37)(23,36)(24,38)(26,40)(27,42)(28,41)(34,43)(35,39)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 879);