Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#352 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #99 |
| commutator | C23 | C2×C4×C11 |
| Frattini, cosocle | C2×D4 | C22×C11 |
| prime radical | C2×D4×C11 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C11 ◁ C2×D4×C11 ◁ G | |
68 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)(14,20)(15,23,24,18)(16,19)(17,21,25,27)(22,26);(12,14,16,22)(13,17,19,25)(15,21,24,27)(18,20,23,26)143 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 156);