Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#430 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | C2×D4 |
| commutator | C4 | C22×C4×C11 |
| Frattini, cosocle | C2×C4 | C23×C11 |
| socle | C2×C11 | C2×C4⋊C4 |
| prime radical | C23×C11 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
102 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)(12,13,16,19,17,20,15,25)(14,21,24,26,22,27,23,18);(12,14,15,23,17,22,16,24)(13,18,19,21,20,26,25,27);(14,22)(18,26)(21,27)(23,24)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 836);