Finite Group DatabasebetaFinite Group Databasebeta| order | 704 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #102 |
| commutator | C8 | C2×C4×C11 |
| Frattini, cosocle | C2×C8 | C22×C11 |
| prime radical | C22×C11 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ G | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,20,17,22,32,21)(14,23,25,36,26,40,31,39)(15,27,29,19,30,37,43,38)(18,28,33,24,35,41,34,42);(12,14,17,26)(13,18,22,35)(15,28,30,41)(16,31,32,25)(19,36,38,39)(20,34,21,33)(23,27,40,37)(24,29,42,43)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(704, 166);