Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×C7×Q8⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C7 | D4 |
| commutator | C4 | C2×C4×C7×C8 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle, prime radical | C23×C7 | C2×C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C7 ◁ C43×C7 ◁ G | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4)(21,22,23,24,25,26,27);(6,9)(7,11)(10,12)(13,14,15,17,16,18,19,20);(5,6,8,10)(7,12,11,9)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1157);