Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#347 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | C42 |
| commutator | C4 | C42×C11 |
| Frattini, cosocle | C2×C8 | C22×C11 |
| socle | C2×C11 | #97 |
| prime radical | C22×C11 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ G | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,18,15,20,22,26,16,19,21,27,23,17,24,25);(13,17,19,20)(14,21)(18,26,27,25)(22,24)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 151);