Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#568 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | C24 |
| commutator | C2 | C23×C4×C11 |
| Frattini, cosocle | C4 | C24×C11 |
| socle | C2×C11 | C23×C4 |
| prime radical | C2×C11×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ G | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,19,16,18,22,25)(14,17,21,24,20,23,26,27);(12,14)(13,17)(15,21)(16,20)(18,23)(19,24)(22,26)(25,27);(14,20)(17,23)(21,26)(24,27);(13,18)(17,23)(19,25)(24,27)374 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 1348);