Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#449 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C11 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ G | |
162 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(17,21)(18,19)(22,23);(16,17)(18,21)(19,23)(20,22);(13,15)(16,18,20,19)(17,21,22,23)209 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 855);