Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C82 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#7015 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C7 | D4 |
| commutator | C4 | C7×C82 |
| Frattini, cosocle | C43 | C22×C7 |
| socle, prime radical | C23×C7 | #95 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C7 ◁ C43×C7 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17,18,20,19,21,22,23);(9,12,14,13)(24,25,26,28,27,29,30,31)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 3);