Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C72 · Schur multiplier (H2) = C2×C7 |
| direct product | C72×#101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C72 | C22⋊C4 |
| commutator | C22 | C2×C4×C72 |
| Frattini, cosocle | C2×C4 | C22×C72 |
| prime radical | C22×C72 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C72 ◁ C2×C4×C72 ◁ G | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 784 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3,4,5,6,7)(15,16,18,22,19,21,27,29)(17,23,24,30,25,20,26,28);(8,9,10,11,12,13,14)(15,17,19,25)(16,20,21,23)(18,26,27,24)(22,30,29,28)539 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 328);