Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C11 · Schur multiplier (H2) = C4 |
| direct product | C11×#322 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C11 | C22 |
| commutator | C2 | C4×C8×C11 |
| Frattini, cosocle | C42 | C22×C11 |
| socle, prime radical | C22×C11 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C11 ◁ G | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,17,15,16,18,19);(13,16)(17,19)(20,21,22,24,23,25,26,27)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 126);