Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#478 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C11 |
| prime radical | C2×D4×C11 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ G | |
102 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)(12,13,15,17)(14,18,19,16)(20,21,22,24)(23,25,27,26);(12,14,15,19)(13,16,17,18)(21,24)(25,26);(21,25)(23,27)(24,26)209 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 884);