Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#398 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C11 | C23 |
| commutator, Frattini, cosocle | C23 | C23×C11 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ G | |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(16,17,18,19)(21,24)(25,26);(13,15)(20,21,23,24)(22,26,27,25);(12,13)(14,15)(16,17)(18,19)(20,22,23,27)(21,25,24,26)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 804);