Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#11267 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C7 | C2×D4 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ C22×C42×C7 ◁ G | |
886 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)(12,13,14,15)(16,17,19,18)(21,26)(22,27)(23,24);(8,9)(10,11)(16,18,19,17)(20,21,24,22,25,27,23,26);(9,11)(17,18)(20,22,23,21,25,26,24,27)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4255);