Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C72 · Schur multiplier (H2) = C22×C7 |
| direct product | C72×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C72 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C72 |
| socle | C2×C72 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C72 ◁ C2×C4×C72 ◁ G | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 784 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3,4,5,6,7)(15,16)(17,24)(18,22)(19,27)(20,23)(21,25)(26,29)(28,30);(8,9,10,11,12,13,14)(15,17)(16,21)(18,25)(19,28)(20,26)(22,29)(23,30)(24,27);(15,18,20,19)(16,22,23,27)(17,25,26,28)(21,29,30,24)686 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 781);