Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C23×C11 |
| prime radical | C11×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ G | |
82 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)(14,22)(15,26)(16,31)(17,19)(18,23)(20,29)(21,28)(24,42)(25,39)(27,37)(30,38)(32,36)(33,41)(34,43)(35,40);(12,14,17,25)(13,18,19,33)(15,27,30,43)(16,32,28,24)(20,36,35,42)(21,37,31,34)(22,38,39,26)(23,40,41,29);(13,19)(18,33)(20,35)(21,31)(22,39)(26,38)(34,37)(36,42)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 916);