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 | C7×C8×#106 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C7×C8 | D4 |
| commutator | C4 | C2×C4×C7×C8 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle, prime radical | C23×C7 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15);(16,17,18,19)(21,24)(22,26)(25,27);(20,21,22,24,23,25,26,27)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1268);