Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C72 · Schur multiplier (H2) = C7 |
| direct product | C72×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C72 | D4 |
| commutator | C4 | C2×C4×C72 |
| Frattini, cosocle | C2×C4 | C22×C72 |
| socle | C2×C72 | C4⋊C4 |
| prime radical | C22×C72 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C72 ◁ C2×C4×C72 ◁ G | |
150 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)(15,16,19,22,20,23,18,28)(17,24,26,29,27,30,25,21);(8,9,10,11,12,13,14)(15,17,18,25,20,27,19,26)(16,21,22,24,23,29,28,30)686 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 335);