Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C4 |
| direct product | C11×#490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C11 |
| prime radical | C11×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ G | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,17)(15,16)(18,19)(21,26)(22,27)(23,24);(13,16)(17,19)(20,21,23,26,25,27,24,22);(12,14,15,18)(13,17,16,19)(20,22,23,21,25,26,24,27)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 896);