Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #120 |
| commutator, Frattini, cosocle | C23 | C23×C11 |
| prime radical | C11×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C23×C11 ◁ G | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,19)(14,17,21,24)(15,18,22,25)(20,26,27,23);(12,14)(13,17)(15,20)(16,21)(18,23)(19,24)(22,27)(25,26);(12,15)(13,18)(14,20)(16,22)(17,23)(19,25)(21,27)(24,26)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 864);